On the Existence for the Plateau Problem in Finite Dimensional Banach Spaces. Ioann VASILYEV Professeur Thierry DE PAUW

THESE` DE DOCTORAT DE l’UNIVERSITE´ SORBONNE PARIS CITE´ Préparéeàl’UniversitéParis Diderot

Ecole´ Doctorale de Sciences Mathématiquesde Paris-Centre (ED 386) L’équipe Géometrieet Dynamique

Titre de la th`ese: On the existence for the Plateau problem in ﬁnite dimensional Banach spaces.

Par Ioann VASILYEV

Th`esede doctorat de Mathematiques.

Dirigéepar: Professeur Thierry DE PAUW. Présentéeet soutenue publiquement àParis le 08 octobre 2018 devant le jury composéede : Thierry De Pauw Prof., Univ. Paris Diderot Directeur de thèse Hervé Pajot Prof., Univ. de Grenoble 1 Rapporteur Robert Hardt Prof., Univ. of Rice Rapporteur 1 Guy David Prof., Univ. Paris Saclay Président du jury Olivier Guedon´ Prof., Univ. Paris-Est Mar.-la-Val. Examinateur Frédéric Helein´ Prof., Univ. Paris Diderot Examinateur Vincent Millot Mdc., Univ. Paris Diderot Examinateur

1Rapporteur non présent le jour de la soutenance. A` ma famille. Résumé

Dans cette thèse,nous nous intéressonsàl’existence de minimiseurs pour le problèmede Plateau. On se place dans le cas des espaces de Banach de dimension finie et dans le contexte des G chaˆınesrectifiables. Nous commen¸conspar améliorerl’un des théorèmesde Busemann, selon lequel chaque hyperplan dans un espace de Banach de dimension finie admet une projection qu’est une contraction pour la mesure d’Hausdorff des sous-ensembles n-1 rectifiables (on les appelle ”good projections”). Nous utilisons cette propriétépour montrer que ces projections n’augmentent pas la masse des chaˆınesrectifiables. On en déduit(en utilisant le théorèmede l’approximation forte) la demi-continuitéinférieurede la masse sur l’espace des G chaˆınesrectifiables. Cela nous permet d’établirl’existence men- tionnéeci-dessus. En plus, nous avons pu prouver que, dans le cadre du problèmeénoncé ci-dessus, il y a toujours une solution dont le support soit àl’intérieurde la combinaison convexe du bord. Nous avons également établiune autre démonstrationde l’existence des G chaˆınesminimisantes. Il s’agit des résultatsqui disent que la mesure d’Hausdorff est égaleàla mesure de Gross et que les mesures de Gross sont semi-continues inférieures.On applique un théorèmede Burago et Ivanov pour démontrer l’inégalitétriangulaire pour des G cycles polyédriquesde dimension deux. Enfin, on l’utilise pour démontrer l’existence des G chaˆınesrectifiables minimisantes de dimension deux. Mots clefs. Problèmede Plateau, Théorie géométriquede la mesure, Calcul des variations. 4 Abstract

In this work we study existence of minimisers for the Plateau problem in case of finite dimensional Banach spaces. We work in the context of polyhedral and rectifiable G chains. We start with improving one theorem by Busemann, which states that each hyperplane in a finite dimensional Banach space admits area non-increasing projections (which by the way we call good projections). Namely, we prove that these projections do not increase Hausdorff measure of arbitrary (n - 1) rectifiable subsets. We use this property in order to show that those good projections do not increase the mass (the one relative to th Hausdorff measure) of (n - 1) rectifiable G chains. From here we derive the lower semi-continuity of the mass on the space of rectifiable G chains (via using the strong approximation theorem). This in turn gives us the desired existence result. Furthermore, we were able to prove that in the problem stated above there always exists a solution with the support inside of the convex hull of the boundary data. Along with the “classical” proof discussed above we have also established another proof of the existence of mass-minimizing (n - 1) rectifiable G chains in finite dimensional Banach spaces. Namely, we have proved that for rectifiable subsets of finite dimensional Banach spaces the Hausdorff and the Gross measures coincide. Since those Gross measures are lower semicontinuous, this provides us with an alternative proof of the existence. We apply one theorem of Burago and Ivanov in our proof of the corresponding triangle inequality for the polyhedral 2-cycles. Finally, we show the existence of mass minimizing 2 rectifiable G chains in finite dimensional Banach spaces. Key words. Plateau problem, Geometric measure theory, Calculus of variations. 6 Remerciements

Je tiens en premier lieu àremercier mon directeur de thèse,Thierry de Pauw. Cher Thierry, tout au long de ces quatre années, j’ai pu apprécierta vision de l’analyse et de la géométrie, ta disponibilitéet surtout ton aide et ton soutien dans les situations compliquéesqui me sont arrivées pendant ces années. Je voudrais te témoignerici toute ma gratitude pour m’avoir guidédurant ces annéeset laisséentrevoir ce que le mot recherche veut dire. Je remercie HervéPajot et Robert Hardt qu’ont acceptéde rapporter ma thèse.Merci tout particulièrement àtoi Herve pour tes questions posées. Ces remarques ont apporté des améliorationssignificatives àmon manuscrit. Sincèresremerciements au Robert Hardt d’avoir lu ma thèse.Cher Bob, merci pour ta lecture trèsattentive et tes remarques. Merci beaucoup àGuy David, Olivier Guédon,Frédéric Héleinet Vincent Millot d’avoir acceptéde faire partie de mon jury. Sincèresremerciements àmes collèguesen particulièreàmes tuteurs: Davide Barilari, Houssein Mourtada, Eric´ Toubiana et Anton Zorich pour leur aide. Je remercie également Nikolai Nikolski, Sergei Kislyakov, Evgueni Abakoumov, Guy David, Laurent Moonens, Vincent Millot et Antoine Lemenant pour les intéressantes dis- cussions mathématiques (et non-mathématiques)que nous avons eues. Merci mille fois àmes amies les plus proches: Antoine Julia, Ratko Drada, Grégoire Sergeant–Perthuis, Emannuel Lecouturier, Siarhei Finski, Vladislav et Daria Stazhinski et Victor Hardy pour leur soutien. J’aimerais exprimer toute ma reconnaissance àtoutes les membres de ma famille qui m’ont soutenu tout au long de ces années et m’ont toujours encouragéàfaire des mathématiques. 8 Contents

R´esum´e ...... 3 Abstract ...... 5 Remerciements ...... 7

1 Introduction (fran¸cais). 11 1.1 L’histoire...... 11 1.2 Discription de la th`ese...... 13

2 Introduction. 17 2.1 Some history of the subject...... 17 2.2 Description of the thesis...... 19 2.3 Structure of the thesis...... 22

3 Rectifiable sets and chains in Banach spaces. 25 3.1 Some preliminary definitions and remarks...... 25 3.2 Some useful theorems on rectifiable sets and chains in Banach spaces. . . . 30

4 Busemann’s theorem and some of its consequences. 33 4.1 Some preliminary deﬁnitions and remarks...... 33 4.2 Busemann’s convexity theorem in codimension one...... 34 4.3 Area non–increasing projections in codimension one...... 39

5 The “classical” proof of the existence of mass minimizing (n−1)–rectiﬁable G chains. 55 5.1 The lower semicontinuity on the space of (n − 1) polyhedral G chains. . . . 55 5.2 The lower semicontinuity on the space of (n − 1)–rectiﬁable G chains. . . . 60 5.3 Solution to the Plateau problem whose support stays inside of the convex hull of the boundary...... 65

6 Gross measures and the second proof of the existence in codimension one. 69 6.1 A little bit more about good projections...... 69 6.2 The Gross measures and another proof of the existence in codimension one. 71 10 CONTENTS

7 The existence in the two dimensional case. 81 7.1 Some preliminary deﬁnitions and remarks...... 81 7.2 Burago–Ivanov’s theorem and some of its consequences...... 82 7.3 The existence for the Plateau problem...... 87

Appendices 91

A Lower semicontinuity and semiellipticity in the RNP spaces. 93 Chapter 1

Introduction (fran¸cais).

1.1 L’histoire.

Résoudrele problèmede Plateau c’est une question trèscélèbrede théoriede la mesure géométriqueet du calcul des variations. Ce problèmeconsiste àtrouver une surface dont le bord coincide avec un countour fixéet qui soit d’aire minimale. Nous soulignons que les mots “surface”, “contour” et “aire” utilisésdans la phrase précédente nécessitent une explication. En effet, il y a plusieurs fa¸conspossibles d’énoncerle problèmede Plateau. Dans le cadre des courants le problèmede Plateau est formuléde la manièresuivante. Soit B un courant (m−1) dimensionnel intégralet rectifiable dont le support soit compact dans Rn et tel que ∂B = 0. Le but est de trouver un courant intégral et rectifiable de dimension m tel que ∂S = B et tel que M(S) ≤ M(T ) pour tout les courants intégrauxrectifiables et qui satisfaisont ∂T = B. Il y a une autre formulation possible du problèmede Plateau. Pour m ∈ [0, . . . , n] soit n n Rm(l2 ,G) le groupe de toutes les chaˆınes m rectifiables dans l’espace euclidien R avec des n coefficients dans un groupe abéliennorméet complet (G, | ... |). Le groupe Rm(l2 ,G) est équipéde la masse M, celle associéeàla mesure de Hausdorff euclidienne. Le problèmede Plateau est le problèmesuivant ( minimiser (T ) M (1.1) n parmi T ∈ Rm(l2 ,G) tel que ∂T = B,

n où B ∈ Rm−1(l2 ,G), spt(B) est compact et ∂B = 0. Grâceaux méthodes de la théoriedes courants normaux et intégrauxet des chaˆınes àcoefficients dans les groupes, pour le cas des espaces euclidiens, le problèmede Plateau admet au moins une solution. Présentons d’abord une description plus détailléede la situation dans le cadre euclidien. L’une des méthodes possibles pour démontrer l’existence est la méthode directe du calcul des variations. Elle consiste en les deux étapes suivantes: la premièreest la semi–continuitéinférieurede la masse par rapport àune topologie bien choisie et la seconde est la compacitéde l’espace des “surfaces” admissibles par rapport à 12 Introduction (fran¸cais). la mêmetopologie. Les topologies les plus fréquemment utiliséessont la topologie plate et la topologie faible étoile. Il existe plusieurs preuves différentes de la semi-continuitéinférieurede la masse dans les espaces euclidiens. La premièreméthode ne fonctionne que dans le cadre des courants. Plus précisément, l’étape principale de cette méthode c’est de démontrer la formule suivante:

m n ∗ M(T ) = sup{T (ω): ω ∈ D (R ), ||ω|| ≤ 1}, où || ... ||∗ est la comasse, ||ω||∗ = sup{|hξ, ωi| : ξ est un m vecteur simple unitaire}. On note qu’une fois que cette formule est vérifiée,la semi-continuitéinférieure par rapport àla topologie faible étoilesuit facilement, car le suprémum de fonctions continues est toujours semi-continue inférieurement. La seconde méthode est baséesur l’observation suivante. Soit W ∈ G(m, Rn) et soit π : Rn → W une projection orthogonale sur W . Alors Lip(π) ≤ 1 et donc π n’augmente pas les mesures Hausdorff des ensembles rectifiables. On en déduitle fait suivant qui ressemble àl’inégalitétriangulaire. Soit P un cycle m polyédrique (i.e. ∂P = 0) tel que PN P = j=1 gj[σj], où σj sont m sont des simplexes qui ne se chevauchent pas. Alors on a que N m X m |g1|H (σ1) ≤ |gj|H (σj). j=2 Cette inégalitédécoule du théorèmede constance. Ensuite, en utilisant les théorèmes d’approximation forte et faible, on peut en déduirela semi-continuitéinférieuresouhaitée. La formule de Crofton fournit encore une méthode applicable pour démontrer que la masse est semi-continue inférieurement. Plus précisément, supposons que A ⊂ Rn soit un ensemble m - rectifiable. Alors, la formule suivante appeléela formule de Crofton (ou la formule intégrale-géométrique)a lieu: Z Z m m −1 H (A) = I1 (A) = dγm(W ) card(πW (y) ∩ A)dy, (1.2)

G(m,Rn) W où πW est la projection orthogonale sur W et γm est une mesure O(m) invariante sur la Grassmannienne G(m, Rn) “bien normalisée”. Grâceàcette formule, on est en mesure de prouver la semi-continuitéinférieurede la manièresuivante : tout d’abord, on obtient facilement que la masse de Hausdorff M coincide avec la masse intégrale-géométrique MI (c’est-à-direavec celle définie par rapport àla mesure intégrale-géométrique)sur l’espace des chaˆınes m - rectifiable G et que la formule suivante est vraie: Z Z M(T ) = MI (T ) = dγm(W ) MhT, πW , yidy.

G(m,Rn) W D’un autre côté,la masse intégrale-géométriqueest toujours semi-continue inférieurement par rapport àla convergence plate. En effet, si Ti → T au sens “plat” avec Ti,T ∈ 1.2 Discription de la thèse. 13

m n R (l2 ,G), alors les morceaux de dimension z´erosatisfont Z F(hTi − T, πW , yi)dy ≤ F(Ti − T ) → 0,

oùl’inégalitéici découle du fait que πW àune constante de Lipschitz egale à1. Donc (peut- êtreen prenant une sous-suite) on aura que F(hTi − T, πW , yi) → 0 pour presque tout y. Enfin, le lemme de Fatou et l’observation que la masse est semi-continue inférieurement sur les morceaux de dimension 0 nous permettent de conclure que MI est semi-continue inférieurement. En ce qui concerne la compacité,l’un des moyens les plus simples de la démontrer consiste en les deux étapes suivantes. La premièreutilise également que les projections sur les ensembles convexes sont 1-Lipschitziennes, c’est àdire la propriététraitéedans les deux derniers paragraphes. Comme ces projecteurs n’augmentent pas la masse, on peut simplement projeter la suite minimisante sur une boule contenant spt(B), voir (1.1). En effet, cette opérationfournit une nouvelle suite minimisante qui maintenant reste dans la boule. La deuxièmeétape concerne le théorèmede compacitéprouvédans [Pau14]. Plus en détail,soit G un groupe localement compact, soit K ⊆ Rn un ensemble compact et soit C > 0 une constante, alors l’ensemble

n Fm(l2 ,G) ∩ {T : N(T ) ≤ C, supp(T ) ⊆ K} est F - compact. La demonstration de ce théorèmeest àson tour baséesur le théorèmede déformationBrian White. On note que cette méthode garantit l’existence d’une solution dont le support est compact.

1.2 Discription de la th`ese.

Malgrécette belle image euclidienne avec plusieurs méthodes applicables décritesdans la section précédente, la situation change lorsque nous prenons un espace de Banach comme espace ambiant. Mêmedans les espaces de Banach de dimension finie, la situation n’est pas assez claire. Il me semble que dans ce dernier cas - qu’on estime de se produire dans les applications - d’autres méthodes devraient êtreimpliquées. En effet, pour les espaces de Banach qui ne possèdent pas de produit scalaire, il n’y a ni dualité,ni formule de Crofton, 3 voir [Sch01]. De plus, il n’est pas difficile de démontrer que dans l’espace X = (R , || ... ||∞) toutes les projections linéaires π : X → W sur l’hyperplan W = {x ∈ X : x + y + z = 0} ont une constante de Lipschitz C strictement supérieureà1. D’autre part, la compacité n’est pas un vrai problèmeici, une fois que l’on accepte des solutions dont le support soit non compact. En effet, un argument diagonal qui ressemble àcelui d’Ambrosio et Schmidt [AS13] pourrait êtreadaptéau contexte des G chaˆınes dans des espaces de Banach de dimension finie. Décrivons plus en détailla variante du problèmede Plateau qui nous intéressera.Sup- posons que (X, || ... ||) est un espace de Banach de dimension n. Nous aimerions souligner 14 Introduction (fran¸cais). ici au tout débutque dans chaque espace de Banach de dimension finie (X, || ... ||) considéré dans cette thèse,on fixe un systèmed’Auerbach e1, . . . , en, de1,..., den, où n est la dimension de X. Cela signifie que e1, . . . , en sont vecteurs unitaires dans X et que de1,..., den sont ∗ vecteurs unitaires dans X tels que pour tout i, j entre 1 et n, dei(ei) = 1 et dei(ej) = 0, une fois que i 6= j. Nous supposons également qu’une structure euclidienne est fixée sur X de telle sorte que e1, . . . , en soit une base euclidienne orthonormée. La norme euclidienne correspondante est notée | ... | et h·, ·i représente le produit scalaire. Notons Rm(X,G), m ∈ [0, . . . , n] le groupe de toutes les chaˆınes m - rectifiables avec des coefficients dans un groupe abelien normécomplet et localement compact (G, | ... |). Equipons le groupe Rm(X,G) de la masse M, celle associéeàla métriquedans X donnéepar la norme || ... ||. Ensuite, le problèmede Plateau correspondant se lit comme suit ( minimiser (T ) M (1.3) parmi T ∈ Rm(X,G) tels que ∂T = B, où B ∈ Rm−1(X,G), spt(B) est compact et ∂B = 0. Le but principal de cette thèseest de démontrer que ce problèmeadmet une solution dans les cas où m = n − 1 et m = 2. Décrivons maintenant nos résultatsprincipaux. Le théorème classique de Busemann [Bus49] montre que, dans tous les espace de Banach de dimension n, la fonction de Hausdorff - Busemann (n − 1) dimensionnelle est convexe. Nous rappelons ce que cela signifie. Soit X un espace Banach. Une fonction φ : GC(m, X) → R, où n > m et GC(m, X) est le cône de Grassmann constituéde m - vecteurs simples dans X, est convexe si elle est égaleàla restriction d’une norme sur Λm(X). La fonction de Hausdorff - Busemann est définiede la fa¸consuivante. Soit B la boule unitaire de X, B = {x ∈ X : ||x|| ≤ 1} et soit G(m, X) la m - Grassmannienne de l’espace X, c’est-à-direl’ensemble des sous-espaces de dimension m de X. Pour 1 ≤ m ≤ n la fonction φBH : G(m, X) → R+ définiecomme α φ (W ) = m BH Hm(B ∩ W ) s’appelle la densitéBusemann - Hausdorff. Alors le théorèmede Busemann dit que la ⊥ ⊥ fonction σB : GC(1,X) → R+ définiepar σB(v) := |v| · φBH (v ), où v est l’hyperplan orthogonal au vecteur v, est convexe. Puis, Busemann dans son article [Bus50] utilise la convexitérévéléeafin de montrer que pour tout hyperplan W ∈ G(n − 1,X), il existe une projection linéaire πW avec deux propriétéssuivantes. Premièrement, πW est une projection sur W et la seconde c’est que

n−1 n−1 H||...||(πW (A)) ≤ H||...||(A) pour toutes ensembles A ⊂ V, où V est un sous-espace linéairede X dont la dimension égale (n−1). Cette deuxièmepropriétésignifie que πW n’augmente pas la mesure Hausdorff des sous-ensembles d’hyperplans. Nous appelons telles projections “W - good”. Dans cette thèse,nous avons démontréque ces projections n’augmentaient pas la mesure de Hausdorff de sous-ensembles (n − 1)-rectifiables dans X. Grâceàces projecteurs, nous 1.2 Discription de la thèse. 15 avons pu imiter la deuxièmeméthode décritedans la section précédente. En particulier, ces projections ont jouéun rôletrèsimportant dans notre démonstrationde l’inégalité triangulaire pour les cycles. Cette méthode a également utiliséles analogues de Banach des théorèmesd’approximation forte et faible. Comme l’on a déjàmentionné,Busemann, dans son article [Bus49] montre que toutes les densitésde Busemann - Hausdorff de dimension n − 1 sont convexes. 50 ans plus tard un nouveau résultatde ce type est apparu. Burago et Ivanov, dans leur article [BI12], montrent qu’un ensemble plat dans un espace de Banach de dimension finie dont la boule est un polyhèdreconvexe possèdela plus petite mesure de Hausdorff parmi toutes les surfaces 2-dimensionnelles ayant le mêmebord. Hélas,en dehors du cas de codimension 1, certaines densitésconvexes peuvent ne permettre aucune contraction pour la mesure de Hausdorff. Néanmoins,en utilisant les résultatsde Burago et Ivanov, nous avons démontréqu’il existe un ”combinaison linéaire”de projections qui est toujours suffisante pour prouver l’inégalité triangulaire pour les 2 cycles polyédriquesavec des coefficients dans un groupe abélienet localement compact. Comme nous l’avons déjàmentionné,il n’existe pas de formule de Crofton dans les espaces de Banach. Malgréce fait, nous avons pu démontrer que dans les espaces de Banach de dimension finie, en codimension un, il existe une version locale de cette formule, qui introduit une nouvelle version des mesures de Gross. Rappelons la définitionde la mesure classique de Hausdorff. Soit S un sous-ensemble d’un espace de Banach de dimension n (X, || ... ||) et soit δ > 0 un nombre réel.Posons pour un entier m tel que 1 ≤ m ≤ n

( ∞ ∞ ) m,δ X m [ H||...||(S) := αm inf ξ (Ai): Ai ⊂ X tel que S ⊆ Ai, diam||...||(Ai) ≤ δ , i=1 i=1

m m m m,δ m 2 où ξ (Ai) = diam||...||(Ai)/2 et αm := π /Γ( 2 + 1). Ensuite, puisque H||...||(S) est monotone décroissant en δ, on peut définir

m m,δ m,δ H||...||(S) := sup H||...||(S) = lim H||...||(S). δ>0 δ→0

m La mesure H||...||(S) est appeléela mesure de Hausdorff m - dimensionnelle. Puis, si πW est une projection W –good, alors la mesure de Gross c’est celle obtenue par la même construction, mais quand on prend la fonction

n−1 n−1 ζ||...||(A) = sup H||...||(πW (A)) W ∈G(n−1,X) au lieu de ξn−1. Bien sûr,nous trichons ici un peu, car pour certains hyperplans, il peut y en avoir plus qu’une seule W - good projection. Cela nous oblige àfaire un bon choix de tels projecteurs. Il s’est avéréque l’ensemble des projections W - good est un fermé.On en déduitqu’il existe un choix universellement mesurable GP : G(n − 1,X) → Hom(X,X) de ces projecteurs. Cette condition de mesurabilitéuniverselle s’est avéréesuffisante pour n−1 prouver que, en codimension 1, la mesure Gross G||...||,GP coincide avec celle de Hausdorff. 16 Introduction (fran¸cais).

Ensuit, il en découleque les deux masses associéescoincident également. Enfin on démontre que la masse de Gross est semi-continue inférieurement. En ce qui concerne la compacitéen codimension un, nous avons pu adapter la méthode qui utilise les projections sur des ensembles convexes, celle décritedans la section précédente. On a remplacéces projections par une composition de plusieurs good projections sur des hyperplans. Ensuite on applique des projections euclidiennes sur des points les plus proches àl’intérieurde ces hyperplans. Il est ànoter que cette approche permet de conclure qu’il existe toujours une solution du problème de Plateau dont le support soit un sous-ensemble de l’enveloppe convexe du bord. Soulignons que, comme conséquence de ce résultat,on donne une réponse (partielle) àune question poséedans l’article [AS13] d’Ambrosio et Schmidt. 3 D’autre part, le cas de l’espace l∞ montre qu’il peut y avoir des chaˆınesminimisantes dont les supports ne restent pas dans conv(spt(B)). Soit u : [0, 1] × [0, 1] → R une fonction 2 Lipschitzienne telle que u = 0 sur ∂([0, 1] ) et satisfaisant Lip∞(u) ≤ 1, oùLip∞ c’est la 2 2 2 constante de Lipschitz par rapport àla métriquede l’espace l∞ . On écrit T0 = E [0, 1] 2 3 et B = ∂T0. Posons Fu : [0, 1] → R : x → (x, u(x)) et T = (Fu)#T0. Alors évidemment ∂T = ∂T0 et il n’est pas trèsdifficile de démontrer que M∞(T ) = M∞(T0) = 1. Dans le cas de dimension 2, on est obligéd’utiliser un autre argument qui ressemble àcelui d’Ambrosio et de Schmidt. Hélas,cette méthode en généralne fournit pas une solution dont le support soit compact. Chapter 2

Introduction.

2.1 Some history of the subject.

Proving the existence and the regularity for the Plateau problem is a paradigmatic question in the Geometric Measure Theory and the Calculus of Variations. It comprises finding a surface of least area spanned by a given closed contour. We emphasise that the words “surface”,“area” and “spanned” utilised in the previous phrase are in need of an explanation. Indeed, there are several possible ways to state the problem. In the context of integral currents the Plateau problem is formulated in the following way. Let B be a compactly supported (m − 1)–dimensional integral rectifiable current in Rn such that ∂B = 0. Then the goal is to find an m–dimensional integral rectifiable current S such that ∂S = B and M(S) ≤ M(T ) for every integral rectifiable current T satisfying ∂T = B. Another possible formulation of the Plateau problem is the following one. For m ∈ n [0, . . . , n] denote by Rm(l2 ,G) the group of all m–rectifiable chains in the Euclidean space Rn with coefficients in a complete locally compact normed Abelian group (G, | ... |). The n group Rm(l2 ,G) is equipped with the mass M, the one relative to the Euclidean Hausdorff measure. The corresponding Plateau problem reads as follows ( minimize (T ) M (2.1) n among all T ∈ Rm(l2 ,G) such that ∂T = B,

n where B ∈ Rm−1(l2 ,G), spt(B) is compact and ∂B = 0. Thanks to the methods of the theory of normal and integral currents and chains with coefficients in groups, for the case of Euclidean spaces the Plateau problem admits at least one solution. Let us first present a more detailed description of the situation in the Euclidean setting. One of the possible methods of proving the existence is the direct method of calculus of variations. It comprises two different steps: the first one is the lower semicontinuity of the mass with respect to some topology and the second one is the compactness of the space of admissible competitors with respect to the same topology. The most frequently used topologies are the flat topology and the weak star topology. 18 Introduction.

There are several diﬀerent proofs of the lower semicontinuity of the mass in Euclidean spaces. The ﬁrst one works in the context of currents. In more details the principal step of this method is in showing the following formula:

m n ∗ M(T ) = sup{T (ω): ω ∈ D (R ), ||ω|| ≤ 1}, where || ... ||∗ stands for the comass, ||ω||∗ = sup{|hξ, ωi| : ξ is a unit simple m vector}. Note that once such formula holds true the lower semicontinuity with respect to the weak star convergence follows easily, since supremum of continuous functions is always lower semicontinuous. The second method is based on the following observation. Let W ∈ G(m, Rn) and let π : Rn → W be an orthogonal projection with the range W . Then Lip(π) ≤ 1 and hence π does not increase Hausdorff measures of rectifiable sets. From this observation one can derive the following fact which resembles the triangle inequality. Let P be a polyhedral m– PN cycle (i.e. ∂P = 0) such that P = j=1 gj[σj], where σj are non–overlapping m–simplexes. Then N m X m |g1|H (σ1) ≤ |gj|H (σj). j=2 It is worth noting that the proof of this inequality requires utilization of the constancy theorem. From here using the strong and the weak approximation theorems one can derive the desired lower semicontinuity. Crofton formula provides yet another applicable method of showing that the mass is lower semicontinuous. More precisely, let A ⊂ Rn be a m–rectifiable set. Then the following formula which is called Crofton formula or L1–integral–geometric formula holds true: Z Z m m −1 H (A) = I1 (A) = dγm(W ) card(πW (y) ∩ A)dy, (2.2)

G(m,Rn) W where πW is the orthogonal projection with the range W and γm is the unique “properly normalized” O(m) invariant measure on the Grassmannian G(m, Rn). With the help of this formula one can prove the lower semicontinuity in the following way. First, as an easy consequence, one readily gets that the Hausdorff mass M coincides with the integral– geometric mass MI (i.e. the one defined with respect to the integral–geometric measure) on the space of m–rectifiable G chains and that the following formula holds true: Z Z M(T ) = MI (T ) = dγm(W ) MhT, πW , yidy.

G(m,Rn) W On the other hand, the integral–geometric mass is lower semicontinuous with respect to m n ﬂat convergence. Indeed, if Ti → T in ﬂat distance with Ti,T ∈ R (l2 ,G), then the zero dimensional slices satisfy Z F(hTi − T, πW , yi)dy ≤ F(Ti − T ) → 0, 2.2 Description of the thesis. 19

where the inequality here follows the fact that πW have Lipschitz constant one. Hence up to a subsequence one has F(hTi − T, πW , yi) → 0 for almost all y. Finally, the Fatou lemma and the observation that the mass is lower semicontinuous on the 0 dimensional slices, allow to conclude that MI is lower semicontinuous. In what concerns the compactness, one of the easiest ways of proving it consists of the following two steps. The ﬁrst step as well requires 1–Lipschitz property of the nearest point projections on convex sets, discussed in the last two paragraphs. Since those projectors do not increase the mass, one can just project the minimizing sequence onto some ball containing spt(B), see (2.1). Indeed, this operation provides a new minimizing sequence which stays inside the ball. The second step relates on the compactness theorem proved in [Pau14]. In more details, this result shows that if G is a locally compact group, K ⊆ Rn is a compact set and C > 0 is a constant, then the set

n Fm(l2 ,G) ∩ {T : N(T ) ≤ C, supp(T ) ⊆ K} is F–compact. The proof of this theorem is in turn based on the deformation theorem for ﬂat G chains by Brian White. It is worth mentioning that this method guarantees the existence of a solution that has compact support.

2.2 Description of the thesis.

Despite of a seemingly nice Euclidean picture with several applicable methods described in the previous section, the situation changes dramatically once we let the ambient space become a Banach space. Even in the finite dimensional Banach spaces the situation is not clear enough. It seems that in the latter case–which is likely to arise in applications–other methods should be involved. Indeed, for Banach spaces that possess no Hilbert structure there exists neither duality, nor Crofton formula see [Sch01]. In addition, it is not difficult 3 to show that in the space X = (R , || ... ||∞) any linear projection π : X → W onto the hyperplane W = {x ∈ X : x + y + z = 0} has Lipschitz constant C strictly greater than 1. On the other hand, the compactness is not a real problem here, once one accepts solutions that have non–compact support. Indeed, a diagonal argument that resembles that of Ambrosio and Schmidt [AS13] can be adapted to the context of G chains in finite dimensional Banach spaces. Let us describe in more details the variant of the Plateau problem we are going to be interested in. Suppose (X, || ... ||) is an n–dimensional Banach space. We would like to emphasize here in the very beginning that in every finite dimensional Banach space (X, || ... ||) considered here, we fix an Auerbach system e1, . . . , en, de1,..., den, where n is the dimension of X. This means that e1, . . . , en are unit vectors in X and de1,...,, den are ∗ unit vectors in X such that for all i, j between 1 and n, dei(ei) = 1 and dei(ej) = 0, once i 6= j. We also suppose that a Euclidean structure is fixed on X in a way that e1, . . . , en is an orthonormal Euclidean basis. The corresponding Euclidean norm is denoted by | ... | and h·, ·i stands for the corresponding scalar product. Denote by Rm(X,G), m ∈ [0, . . . , n] the group of all m–rectifiable chains with coefficients in a complete locally compact normed 20 Introduction.

Abelian group (G, | ... |). Equip the group Rm(X,G) with the mass M, the one relative to the metric in X induced by the norm || ... ||. Then the corresponding Plateau problem reads as follows ( minimize (T ) M (2.3) among all T ∈ Rm(X,G) such that ∂T = B, where B ∈ Rm−1(X,G), spt(B) is compact and ∂B = 0. The main goal of this thesis is to show that this problem admits a solution in case when m = n − 1 and m = 2. Let us next describe and motivate our main results. The classical theorem by Buse- mann [Bus49] shows that in any n–dimensional Banach space X the (n − 1) dimensional Hausdorff–Busemann area function is extendibly convex. We remind to the reader of what this means. Let X be a Banach space. A function φ : GC(m, X) → R, where n > m and GC(m, X) is the Grassmann cone of simple m–vectors in X, is extendibly convex if it is the restriction of some norm on Λm(X). The Hausdorff–Busemann area function in X is defined as follows. Let B be the unit ball of X, B = {x ∈ X : ||x|| ≤ 1} and denote by G(m, X) the m–Grassmannian of the space X, i.e. the set of all m–dimensional planes in X. For 1 ≤ m ≤ n the function φBH : G(m, X) → R+ defined as α φ (W ) = m BH Hm(B ∩ W ) is called the Busemann–Hausdorff density. Then Busemann’s theorem tells that the func- ⊥ ⊥ tion σB : GC(1,X) → R+ defined as σB(v) := |v| · φBH (v ), where v is the hyperplane orthogonal to the vector v, is extendibly convex. In addition, Busemann in the paper [Bus50] used the revealed convexity in order to show that for any hyperplane W ∈ G(n − 1,X) there exists a linear projection πW with the following two properties. First, it has range W and second is that

n−1 n−1 H||...||(πW (A)) ≤ H||...||(A) for any set A ⊂ V, where V is some (n − 1)–dimensional linear subspace of X. The second property means that πW does not increase the Hausdorff measure of the subsets of hyperplanes. We call such projections W –good. In this thesis we have shown that these projections do not increase Hausdorff measure of arbitrary (n − 1)–rectifiable subsets of X. With help of these projectors we were able to mimic the second method described in the previous section. In particular those projections played crucial role in our proof of the corresponding version of the triangle inequality for cycles. This method has also required Banach space analogues of the strong and the weak approximation theorems. As it was already mentioned, Busemann in his paper [Bus49] shows that any (n − 1)– dimensional Busemann–Hausdorff density is convex. It was only 50 years later, when a new result of such kind appeared. Namely, in their paper [BI12], Burago and Ivanov show that a bounded planar region in a finite dimensional Banach space whose unit ball is a convex polyhedron, has the least two–dimensional Hausdorff measure among all compact smooth two-dimensional surfaces having the same boundary. Alas, outside of the codimension 2.2 Description of the thesis. 21 one case, some extendibly convex densities can happen to admit no area non–increasing projections. Nevertheless, using the result of Burago and Ivanov we showed that there exists a “linear combination” of projections that is still enough to prove the triangle inequality for polyhedral 2–cycles with coefficients in arbitrary normed locally compact Abelian group. As we have already mentioned there exists no Crofton formula in Banach spaces. De- spite of this fact, we were able to show that in finite dimensional Banach spaces in the codimension one case, there exists a “local” version of this formula, which brings in a new version of Gross measures. Let us present a more detailed description of those. Recall the definition of the classical Hausdorff measure. Let S be any subset of an n–dimensional Banach space (X, || ... ||) and let δ > 0 be a real number. For an integer m such that 1 ≤ m ≤ n we define

( ∞ ∞ ) m,δ X m [ H||...||(S) := αm inf ξ (Ai): Ai ⊂ X such that S ⊆ Ai, diam||...||(Ai) ≤ δ , i=1 i=1

m m m m,δ m 2 where ξ (Ai) = diam||...||(Ai)/2 and αm := π /Γ( 2 + 1). Next, since H||...||(S) is monotone decreasing in δ, we are allowed to deﬁne

m m,δ m,δ H||...||(S) := sup H||...||(S) = lim H||...||(S). δ>0 δ→0

m The measure H||...||(S) is called the m–dimensional Hausdorﬀ measure. Now, if πW is a W –good projection, then the Gross measure is the one obtained from the very same construction, but with the function

n−1 n−1 ζ||...||(A) = sup H||...||(πW (A)) W ∈G(n−1,X) used as a gauge instead of ξn−1. Of course, here we cheat a little bit, since for certain hyperplanes there might be more than one W –good projection. This in turn forces one to make a proper choice of those projectors. It turned out that the set of W –good projections is closed. This observation further allowed us to prove that there exists a universally measurable choice GP : G(n − 1,X) → Hom(X,X) of those projectors. This universal measurability condition turned out to be suﬃcient to prove that in codimension one the n−1 Gross measure G||...||,GP coincides with the Hausdorﬀ measure, which in turn implies that the corresponding masses coincide as well. The Gross mass was further shown by us to be lower semicontinuous. In what concerns the compactness in the codimension one case we were able to adapt the method that requires projections onto convex sets, described in the previous section. We replace these projections with a superposition of a number of good projections onto hyperplanes and further apply Euclidean nearest point projections inside these hyperplanes. It is worth mentioning that this approach allows to conclude that there always exists a solution of the Plateau problem whose support is a subset of the convex hull of the boundary. We would like to emphasize that as a consequence of this result, we give a (partial) answer to a question raised in the paper [AS13] by Ambrosio and Schmidt. On the other hand, a 22 Introduction.

3 simple example of the space l∞ shows that there also exist mass minimizing chains whose supports need not to stay inside of conv(spt(B)). Indeed let u : [0, 1] × [0, 1] → R be any 2 Lipschitz function such that u = 0 on ∂([0, 1] ) and satisfying Lip∞(u) ≤ 1, where Lip∞ 3 2 2 signifies the Lipschitz constant with respect to the l∞ metric. Write T0 = E [0, 1] and 2 3 B = ∂T0. Define Fu : [0, 1] → R : x → (x, u(x)) and T = (Fu)#T0. Then obviously ∂T = ∂T0 and it is not difficult to show that M∞(T ) = M∞(T0) = 1. In the two–dimensional case we have to use another argument which resembles that of Ambrosio and Schmidt. Alas, this method in general does not provide a compact support solution.

2.3 Structure of the thesis.

Here we describe the composition of the chapters of the thesis. In the second chapter we first introduce the main definitions used in the manuscript. Those definitions one can divide into two parts: those related to areas and volumes of sets in Banach spaces and those connected with rectifiable chains with coefficients in groups. We further cite in the second part of the chapter a number of classical results concerning rectifiable sets and chains with coefficients in groups. Among them are coarea formula, B.Kirchheim’s theorem, constancy theorem, Eilenberg’s inequality, Besicovitch covering theorem and compactness theorem for flat chains. In the third chapter we first give a careful proof (Theorem 4.2.1) of Busemann’s theorem on the convexity of the Hausdorff–Busemann area function. Then we prove Theorem 4.3.2 which shows that there each hyperplane admits area non–increasing projections. Next we derive in Theorem 4.3.4 the fact that these projectors do not increase Hausdorff measure of (n − 1)–rectifiable subsets. Finally in this chapter we prove Theorem 4.3.5 which says that good projections do not increase the mass of (n − 1) polyhedral G chains. The forth chapter begins with establishing Theorem 5.1.1 where we prove the validity of the “triangle inequality” disclosed before in the codimesion one case. Then we prove our Theorem 5.1.2 where we give a necessary and sufficient condition for the lower semicontinuity of the mass on the space of polyhedral chains in finite dimensional Banach spaces. This condition is nothing but the “triangle inequality”. We further derive in Theorem 5.2.1 the lower semicontinuity of the mass on the space of G rectifiable chains via using the strong approximation Theorem 5.2.2. Finally in this chapter we prove our Theorem 5.3.1 where the existence of mass minimising (n − 1)–rectifiable G chains with compact support is established. The fifth chapter consists of the following three principal results. The first one is Proposition 6.1.1 where we prove that the set of good projections is closed and with help of which we further derive the fact that there exists a universally measurable selection of n−1 n−1 good projections. Later in Theorem 6.2.1 we show that H||...||(A) = G||...||,GP(A) for (n − 1) rectifiable subsets of X. Theorem 6.2.2, the last result of this chapter, is the one in which we show that the Gross mass MG is lower semicontinuous with respect to flat norm. We begin the sixth chapter with a proof of the result of Burago and Ivanov mentioned in 2.3 Structure of the thesis. 23 the previous section, our Theorem 7.2.2. We further apply this theorem in our proof of the corresponding triangle inequality for the polyhedral 2–cycles, which is our Theorem 7.3.1. In the last part of this chapter, in Theorem 7.3.2, we prove the corresponding existence result, which in turn uses a compactness argument that resembles that of Ambrosio and Schmidt. 24 Introduction. Chapter 3

Rectiﬁable sets and chains in Banach spaces.

3.1 Some preliminary deﬁnitions and remarks.

We begin with a number of useful deﬁnitions and remarks.

Deﬁnition 3.1.1. (Norm) Let X be a real linear space. We call a function || ... || : X → R a norm if

1. for every x ∈ X one has ||x|| ≥ 0, and ||x|| = 0 if and only if x = 0,

2. for every x ∈ X and α ∈ R one has ||αx|| = |α|||x||,

3. for all x, y ∈ X one has ||x + y|| ≤ ||x|| + ||y||.

We shall also call the space (X, || ... ||) a Banach space.

Deﬁnition 3.1.2. (Supporting hyperplane) Let X be a real linear space of dimension n, let S ⊂ X be an arbitrary subset and let x ∈ S. An aﬃne n − 1 dimensional subspace W ⊂ X is called a supporting hyperplane of S at x if

1. x ∈ W,

2. S is entirely contained in one of the two closed half–spaces bounded by the W .

Remark 3.1.1. Let X be a Banach spaces and let || ... || be a norm on X. Throughout this thesis B||...||(x, r) with x ∈ X and r > 0 signiﬁes the closed ball or radius r centered at the point x whereas U||...||(x, r) stands for the corresponding open ball.

Definition 3.1.3. (Diameter) For any subset S ⊂ X of a Banach space (X, || ... ||) define its diameter as diam(S) = sup{||x − y|| : x, y ∈ S}. 26 Rectifiable sets and chains in Banach spaces.

Definition 3.1.4. (Hausdorff measure) Let S be any subset of an n–dimensional Banach space (X, || ... ||) and let δ > 0 be a real number. For an integer m such that 1 ≤ m ≤ n we define

( ∞ m ∞ ) X diam(Ai) [ Hm,δ (S) := α inf : A ⊂ X such that S ⊆ A , diam(A ) ≤ δ , ||...|| m 2 i i i i=1 i=1

m m 2 m m where αm := π /Γ( 2 + 1). By definition αm = H2 (B), where B = {x ∈ R : |x|2 ≤ 1} m and H2 is any Euclidean Hausdorff measure. m,δ Next, since H||...||(S) is monotone decreasing in δ, we are allowed to define

m m,δ m,δ H||...||(S) := sup H||...||(S) = lim H||...||(S). δ>0 δ→0

m The measure H||...||(S) is called the m–dimensional Hausdorff measure. Remark 3.1.2. In every real linear space X of dimension n considered in this thesis, we fix an Auerbach system e1, . . . , en, de1,..., den, where n is the dimension of X. By this we ∗ mean that e1, . . . , en are unit vectors in X and de1,...,, den are unit vectors in X such that for all i, j between 1 and n, dei(ei) = 1 and dei(ej) = 0, once i 6= j. We also fix a Euclidean structure on X defined in a way that e1, . . . , en is an orthonormal Euclidean basis. We denote by | ... | the corresponding Euclidean norm and by h·, ·i the corresponding scalar product.

Deﬁnition 3.1.5. (Dyadic semi–cubes) Let X as in the previous remark. Let k ∈ Z, deﬁne −k Qk, a system of dyadic semi–cubes of edge 2 as the family of all cubes Q that possess the following form

−k −k Q = {x ∈ X: for all 1 ≤ j ≤ n, 2 mj ≤ hx, eji < 2 (mj + 1), for some m1, . . . , mn ∈ Z}.

Remark 3.1.3. We shall write Hm for the Hausdorﬀ measure, deﬁned with respect to the Euclidean norm of X, described in the previous Remark.

n−1 n−1 Remark 3.1.4. Note that the measures H and H||...|| are both Haar invariant measures. Hence, according to [Mat99] Theorem 3.1 they are proportional.

Deﬁnition 3.1.6. (Exterior algebra) The k–th exterior algebra of a vector space X, denoted as Λk(X), is the vector space spanned by elements of the form x1 ∧ x2 ∧ · · · ∧ xk, where ∧ stands for the exterior product (contact [Fed69], 1.3.1) and xi ∈ X, i = 1, 2, . . . , k. ∗ Let v1, . . . , vn ∈ X and w1, . . . , wn ∈ X , deﬁne the parity as

n hv1 ∧ ... ∧ vn, w1 ∧ ... ∧ wni = det{hvi, wji}i,j=1.

Deﬁnition 3.1.7. (Grassmannian) The k–th Grassmannian of a vector space X denoted G(k, X) is the set of all of m–dimensional linear subspaces of X. 3.1 Some preliminary deﬁnitions and remarks. 27

Definition 3.1.8. (Grassmann cone) The k–th Grassmann cone of a vector space X, denoted GCk(X), is defined as follows: GCk(X) = {ξ ∈ Λk(X): ξ is a simple vector}. Recall that ξ ∈ Λk(X) is a simple multi–vector if ξ = v1 ∧ ... ∧ vk for some v1, . . . , vk ∈ X. Definition 3.1.9. (Rectifiable sets) Let X be a metric space. We call a set E ⊆ X m– m rectifiable if there exists a countable set of Lipschitz functions fi : Ai → X where Ai ⊂ R are bounded subsets such that ∞ ! m [ H E\ fi(Ai) = 0. i=1 Definition 3.1.10. (Approximate tangent spaces) Let M ⊂ X be a subset of a finite dimensional Banach space X that is measurable with respect to the m–dimensional Hausdorff measure, and such that the restriction measure Hm M is a Radon measure. We say that an m–dimensional subspace P ∈ G(m, X) is an m–approximate tangent space to M at a certain point x, denoted Tanm(M, x) (and sometimes TxM, namely when there is no need to specify m) if m m (H M)x,λ * H P weakly in the sense of measures. Here for any measure µ we denote by µx,λ the rescaled and translated measure, −n µx,λ(A) := λ µ(x + λA). Remark 3.1.5. Let M,X and x be as in the previous definition. If an approximate tangent plane P to M at x exists, then it is unique. This follows from [Mat99], Theorem 15.19. Definition 3.1.11. (Densities of sets) Let (X, || ... ||) be an n–dimensional Banach space and let 0 ≤ s ≤ n be a real number. Suppose that E is a Hs–measurable set in X. The upper and lower Lebesbue s–density of E at x ∈ X are defined by s s,∗ H (B(x, r) ∩ E) Θ (E, x) = lim sup s , r→0 αsr s s H (B(x, r) ∩ E) Θ∗(E, x) = lim inf s . r→0 αsr If they agree the common value s s s,∗ Θ (E, x) = Θ∗(E, x) = Θ (E, x) is called the s–density of the set E at x. Definition 3.1.12. (Polyhedral chains) Let (X, || ... ||) be an n–dimensional Banach space and let G be a complete normed Abelian group. We define Pm(X,G), the Abelian group of m polyhedral chains, as the group of equivalence classes of formal sums of the elements of type N X P = gj[σj], j=1 where gj ∈ G and σj are oriented m–simplexes. 28 Rectifiable sets and chains in Banach spaces.

Deﬁnition 3.1.13. (Top–dimensional chains) Let (X, || ... ||) be an n–dimensional Banach space and let G be a complete normed Abelian group. The group Pn(X,G) is called the group of top–dimensional polyhedral chains in X.

We adopt the following conventions: if [σ] is a simplex endowed with the canonical orientation then we endow −[σ] with the opposite orientation. Moreover g(−[σ]) = (−g)[σ] = −g[σ]; finally any summand whose coefficient is the neutral element of G may be omitted P since it gives null contribution. It follows that P = gi[σi] is the identity element if and only if every gi is the neutral element of G, and the inverse element is obtained by P −P = −gi[σi], thus the group is well defined. We say that two simplexes σ1 and σ2 are non–overlapping if int(σ1) ∩ int(σ2) = ∅. We S define the support spt(P ) of P as spt(P ) = j σj, once σi are non–overlapping and the corresponding elements gi 6= 0.

Definition 3.1.14. (Boundary of polyhedral chains) Define a group homeomorphism ∂ : P Pm(X,G) → Pm−1(X,G) called boundary as follows. Let P = j gj[σj]. We first define its value on the basis elements i.e. on simplixes σj and then extend it by the formula X ∂P = gj∂[σj]. j

We require ∂[σ] to be the sum of all faces of σ each one endowed with an orientation compatible to the exterior normal vector of σ.

Deﬁnition 3.1.15. (Mass of polyhedral chains) Let (X, || ... ||) be an n–dimensional Ba- nach space and G be a complete normed Abelian group. Let P ∈ Pm(X,G) be a polyhedral PN m chain such that P = j=1 gj[σj], where σj are non–overlapping. We deﬁne the mass M||...|| of P , as N X m M||...||(P ) = |gj|H||...||(σj). j=1

Remark 3.1.6. We shall often drop the sign || ... || and simply write M for the mass instead of M||...||.

Remark 3.1.7. The mass M is well deﬁned.

Definition 3.1.16. (Flat norm) Define flat norm F(P ) of a polyhderal chain P ∈ Pm(X,G) as follows

F(P ) = inf{M(Q) + M(R): Q ∈ Pm(X,G),R ∈ Pm+1(X,G),P = Q + ∂R}.

Definition 3.1.17. (Flat chains) Let (X, || ... ||) be an n–dimensional Banach space and G be a complete normed Abelian group. Define the group of the m–dimensional flat chains denoted by Fm(X,G), as the F completion of Pm(X,G). 3.1 Some preliminary definitions and remarks. 29

Deﬁnition 3.1.18. (Lipschitz chains) Let (X, || ... ||) be an n–dimensional Banach space and G be a complete normed Abelian group. We deﬁne Lm(X,G), the Abelian group of m–Lipschitz chains, as the subgroup of Fm(X,G) formed by the elements of type

N X P = gjγj#[σj], j=1

m where gj ∈ G, σj ⊂ R are oriented m–simplexes and γj : σj → X are Lipschitz mappings. Definition 3.1.19. (Rectifiable chains) Let (X, || ... ||) be an n–dimensional Banach space and G be a complete normed Abelian group. Define the group of the m–dimensional rectifiable chains denoted by Rm(X,G), as the M completion of Lm(X,G). Remark 3.1.8. The support, the restriction and the boundary of rectifiable chains are well defined, recall [PH14]. We shall utilize the symbol for the restrictions.

Remark 3.1.9. Each T ∈ Rm(X,G) is defined by an m–rectifiable set A, which we will call set(T ) and a Hm–integrable G–valued multiplicity function which is often denoted by θT . We shall often write m T = θT H set(T ). S S If set(T ) = γ ( i Ai) with Ai mutually disjoint and where γ : i Ai → X is Lipschitz, then we shall also write T = [γ, Ai, g]. If θT is a constant, say θT ≡ g ∈ G, then we simply write T = g[set(T )]. Definition 3.1.20. (Slicing) Let (X, || ... ||) be a Banach space of dimension n. If T ∈ k Rm(X,G),T = [γ, Ai, g] and f : X → R is a Lipschitz mapping, then slicing (or slice) k hT, f, yi ∈ Rm−k(X,G) is defined for H almost every y as

m−k −1 hT, f, yi = ±θT H (f (y) ∩ set(T )), where the sign ± depends on the behaviour of f around the point y, see [PH14]. Note that this is well defined, since the sets f −1(y) ∩ set(T ) are (m − k)–rectifiable for Hk almost all y, according to [Fed69], 3.2.22. Proposition 3.1.1. (see [PH12]) Let (X, || ... ||) and (Y, ||| ... |||) be two finite dimensional Banach spaces of dimensions n and d respectively and let G be a complete normed Abelian group. Suppose that S,T ∈ Rm(X,G) and f, φ : X → Y are Lipschitz functions, and U ⊂ X is a subset of X. Then

• M(S + T ) ≤ M(S) + M(T ), • M(T U) ≤ M(T ),

m • M(φ#T ) ≤ Lip(φ) M(T ), R d • M(hT, f, yi)dy ≤ Lip(f) M(T ). Y 30 Rectiﬁable sets and chains in Banach spaces.

3.2 Some useful theorems on rectiﬁable sets and chains in Banach spaces.

For the reader’s convenience we gather in this section most of theorems on rectiﬁable sets and chains in Banach spaces cited in this thesis. First we state the following theorem which is called weak approximation theorem. This result is proved in [Pau14].

Theorem 3.2.1. Assume (Y, || ... ||) is a ﬁnite dimensional Banach space , T ∈ Rn(Y,G) is so that ∂T ∈ Rn−1(Y,G) and spt(T ) is compact and let ε > 0. Then there exists P ∈ Pn(Y,G) such that

• M(P ) ≤ ε + M(T ),

• M(∂P ) ≤ ε + M(∂T ), •F (T − P ) ≤ ε,

• spt(P ) ⊆ B(spt(T ), ε), where B(spt(T ), ε) = {y ∈ Y : ||y − x|| ≤ ε for some x ∈ spt(T )} is the ε–neighborhood of the set spt(T ). Second we present a constancy theorem for G chains in finite dimensional Banach spaces. The proof of this result can be found in [PH14]. Theorem 3.2.2. Let Y be an m–dimensional Banach space and let G be a complete normed Abelian group. If T ∈ Fm(Y,G),U is a connected open subset of Y, and (∂T ) U = 0, then T U = g[U] for some g ∈ G, see Remark 3.1.9. The following theorem by B.Kirchheim [Kir94] shows that rectifiable subsets of a metric spaces have density equal to 1 almost everywhere. Theorem 3.2.3. Let X be a metric space and let A be an m–rectifiable subset of X. Then for Hm almost all x ∈ A there holds Hm(A ∩ B(x, r)) lim m = 1. r→0 αmr We proceed to the Eilenberg inequality. We give a version borrowed from the book [BZ88]. Theorem 3.2.4. Let X and Y be separable metric spaces and let f : X → Y be a Lipschitz mapping. Then for any A ⊆ X and 1 ≤ m ≤ n there holds

∗ Z α α Hn−m(A ∩ f −1(y))dHm(y) ≤ m n−m (Lip(f))n Hn(A), αn where R ∗ stands for the upper Lebesgue integral. 3.2 Some useful theorems on rectiﬁable sets and chains in Banach spaces. 31 Next we state the area formula for Lipschitz mappings between Euclidean spaces. Our n version is the one proved in the book [EG15]. Recall that for a Lipschitz mapping f : l2 → m n n l2 , n ≤ m for H almost all x ∈ l2 its Jacobian J f is given by the following formula

J f(x) = pdet Df T (x) · Df(x), where Df(x) signifies the differential matrix of the mapping f at the point x. Indeed, this definition makes sense since by Rademacher’s theorem f is differentiable for Hn almost all n x ∈ l2 .

n m Theorem 3.2.5. Let f : l2 → l2 be Lipschitz continuous, n ≤ m. Then for each Lebesgue measurable subset A ⊂ Rn, Z Z J f(x)dx = card(A ∩ f −1(y))dy.

A Rm The following theorem guarantees the almost everywhere existence of approximate tangent spaces of rectiﬁable sets. One can consult the book [Mat99] for its proof.

Theorem 3.2.6. Suppose E ⊂ Rn is an Hm measurable and m–rectiﬁable set with m m H (E) < ∞. Then for H almost every x ∈ E the approximate tangent space Tanm(E, x) exists, is unique and moreover one has Θm(E, x) = 1. Compactness theorem from [Pau14] that we are about to state will play crucial role in our existence theorem in the 4–th chapter. Theorem 3.2.7. If X is a ﬁnite dimensional Banach space, G is a locally compact normed Abelian group, K ⊆ X is a compact set and C > 0 is a constant, then the set

Fm(X,G) ∩ {T : M2(T ) + M2(∂(T )) ≤ C, supp(T ) ⊆ K}, where M2 stands for the Euclidean mass, is F–compact. The following result by Kirszbraun says that we can extend Lipschitz mappings to the whole space.

n Theorem 3.2.8. Any Lipschitz mapping from a subset of l2 can be extended to a Lipschitz n mapping defined on the whole l2 with the same Lipschitz constant. We finally present the Besicovitch covering theorem. Theorem 3.2.9. Let µ be a Borel regular measure on a finite dimensional Banach space X and let A ⊂ X be a µ–measurable subset such that µ(A) < ∞. Suppose F is a collection of closed balls such that inf{r : B(a, r) ∈ F } = 0 for all a ∈ A. Then there exists a countable subcollection disjoint of F that covers µ almost all of A. The proof of this result can be found in [Fed69] or in [Mat99], Theorem 2.7. 32 Rectifiable sets and chains in Banach spaces. Chapter 4

Busemann’s theorem and some of its consequences.

Technical by their nature, the results proved in this section will play a crucial role throughout this thesis.

4.1 Some preliminary deﬁnitions and remarks.

Let us recall a number of useful definitions and remarks. We first define a density and a volume density. Definition 4.1.1. (Density and volume density) Let (X, || ... ||) be an n–dimensional Banach space and let 1 ≤ m ≤ n. An m–density function is a continuous function φ : G(m, X) → R+ that is positively homogenous of degree one. An m–density φ is an m–volume density function if for all a ∈ G(m, X) one has φ(a) ≥ 0 with equality if and only if a = 0. Remark 4.1.1. Let X and m be as in the previous definition and let φ be a m–volume density function on X. Then, for a m–rectifiable set S ⊂ X one defines the corresponding volume Volφ(S) as Z m Volφ(S) := φ(TxS)dH (x), S where TxS stands for the corrseponding approximate tangent m–plane, see 3.1.10. We proceed with a very natural example of a volume density. Definition 4.1.2. (Busemann–Hausdorff density) Let (X, || ... ||) be an n–dimensional Banach space with the unit ball B = {x ∈ X : ||x|| ≤ 1}. For 1 ≤ m ≤ n we define the Busemann–Hausdorff density function φBH : G(m, X) → R+ as α φ (W ) = m , BH Hm(B ∩ W ) 34 Busemann’s theorem and some of its consequences. where W ∈ G(m, X). We further define one function which is a slight modification of the n − 1 dimensional Busemann–Hausdorff density. This function will be frequently used in this chapter.

Definition 4.1.3. (Function σB) Let (X, || ... ||) be an n–dimensional Banach space with the unit ball B = {x ∈ X : ||x|| ≤ 1}. Define a function σB : X → R+ as follows |f| σ (f) := α = φ (f ⊥)|f| , B n−1 Hn−1(B ∩ f ⊥) BH where by f ⊥ we denote the hyperplane orthogonal to the vector f. Let us give two examples of Busemann–Hausdorff functions.

2 Remark 4.1.2. In case when X = (R , l2), it is obvious that

{x ∈ X : σB(x) = 1} = {x ∈ X : ||x||2 = 1}.

2 Remark 4.1.3. In case when X = (R , l∞), direct calculation shows that

{x ∈ X : σB(x) = 1} = {x ∈ X : ||x||∞ = 1}. The following property of densities is going to be very important for us. Deﬁnition 4.1.4. (Extendibly convex densities) A k–volume density function φ on an n– dimensional Banach space X, n > k is said to be extendibly convex if it is the restriction of a norm on Λn(X) to the cone of simple k–vectors in X. Remark 4.1.4. Note that in case when m = n − 1 the notion of extendibly convex density coincides with the usual convexity.

4.2 Busemann’s convexity theorem in codimension one.

We recall a classical result concerning sections of convex sets in ﬁnite dimensional Banach spaces. This result is a theorem ﬁrst proved by Herbert Busemann. The proof presented here is taken from the book by Thompson [Tho96]. Theorem 4.2.1. (Busemann) Let (X, || ... ||) be an n–dimensional Banach space with the unit ball B = {x ∈ X : ||x|| ≤ 1}. Then the function σB is convex.

Proof. Since σB is a positively homogeneous function of degree one, we need only to show that it is subadditive. Take two vectors f1, f2 ∈ X. Deﬁne by f3 their sum: f3 = f1 + f2, and by L the two–dimensional subspace of X generated by f1 and f2,L := span{f1, f2}. Let M denote the orthogonal complement of L and let Mi, i = 1, 2, 3 be the linear (n − 1)– dimensional subspace generated by M and fi,Mi := span{M, fi}. Let φ1, φ2 ≥ 0 and let

fi gi = φi , i = 1, 2. |fi| 4.2 Busemann’s convexity theorem in codimension one. 35

Deﬁne vector g3 as the intersection of the segment [g1, g2] and the ray from 0 through f3 (hence g3 := φ3f3/|f3| for some real number φ3(= φ3(φ1, φ2)) ≥ 0). We introduce two more abbreviations concerning areas of slices of B, let n−2 fi ρi(φ) := H B ∩ M + φ |fi| R ∞ for φ ≥ 0 and i = 1, 2, 3, and let Vi := 0 ρi(φ)dφ. We collect most of the introduced notations in the following picture:

n−1 Note that the functions ρi are integrable, and that Vi = H (B ∩Mi). Both statements follow from the Fubini theorem. We claim that the functions ρi are continuous. Indeed, n−2 fi ρi(φ) = H (B ∩ Mi) ∩ M + φ |fi| since (M + φfi/|fi|) ⊂ Mi. Furthermore the set B ∩ Mi is convex and lies in the (n − 1)– dimensional space Mi. Hence by Brunn’s lemma 4.2.1 each function ρi is continuous as a composition of two continuous functions, and our claim follows. 36 Busemann’s theorem and some of its consequences.

Throughout this thesis the symbol 4 stands for a triangle. We will ﬁrst show that |f | |f | |f | 1 + 2 ≥ 3 . (4.1) V1 V2 V3

We start with calculating the coeﬃcient φ3 = φ3(φ1, φ2) in two diﬀerent ways. Comparing the areas of the triangles 40g1g2, 40g1g3 and 40g2g3 gives

φ1φ2 sin(g10g2) = φ1φ3 sin(g10g3) + φ2φ3 sin(g20g3), and hence sin(g 0g ) sin(g 0g ) sin(g 0g ) 1 2 = 2 3 + 1 3 . (4.2) φ3 φ1 φ2

Likewise, using the law of sines for the triangles 40f1f2, 40f1f3 and 40f2f3 we conclude that sin(f 0f ) sin(f 0f ) sin(f 0f ) 1 2 = 1 3 = 2 3 , |f3| |f2| |f1| and hence, thanks to the line (4.2), we infer that |f | |f | |f | 3 = 1 + 2 . (4.3) φ3 φ1 φ2

On the other hand, expressing g3 as a convex combination of g1 and g2, we get that g3 = αg1 + (1 − α)g2 for some α ∈ (0, 1). Hence

φ3 φ1 φ2 f3 = α f1 + (1 − α) f2. |f3| |f1| |f2|

Comparing this representation of f3 with the deﬁnition of f3 we conclude that φ |f | φ |f | α = 3 1 , (1 − α) = 3 2 . (4.4) |f3| φ1 |f3| φ2

As a direct consequence of the deﬁnitions of g1, g2 and g3 and of the convexity of B, we infer that B ∩ (M + g3) ⊇ α[B ∩ (M + g1)] + (1 − α)[B ∩ (M + g2)]. Hence also n−2 n−2 ρ3(φ3) = H (B ∩ (M + g3)) ≥ H (α[B ∩ (M + g1)] + (1 − α)[B ∩ (M + g2)]) . (4.5) We will need the following lemma, which we present here with a proof. This lemma is sometimes called Brunn’s theorem. Lemma 4.2.1. (Brunn) Let (X, || ... ||) be an n–dimensional Banach space and let K ⊂ X be a convex body. Deﬁne its parallel slice Kt as an intersection of K with the hyperplane n X Wt = {x ∈ X : x = λiei, λ1 = t}, i=1 i.e. Kt := K ∩ Wt. Then the function 1 n−1 n−1 t → H (Kt) is convex. 4.2 Busemann’s convexity theorem in codimension one. 37

Proof. Let s, r, t ∈ R with s = (1 − λ)r + λt for some λ ∈ (0, 1), and let Ks,Kr,Kt be (n − 1)–dimensional parallel slices of K, as in the formulation of Lemma 4.2.1. Project by means of the orthogonal projection π the sets Ks,Kr and Kt onto the hyperplane Wt. We claim that the set π(Ks) contains the set λπ(Kr) + (1 − λ)π(Kt). To show this, take two −1 points x ∈ π(Kr) ⊂ Wt and y ∈ π(Kt) ⊂ Wt, connect the pointsx ˜ = (r, x) = π (x) ∩ Wr −1 andy ˜ = (t, y) = π (y) ∩ Wt by a straight line. Note that since K is convex, the segment of this line between the pointsx ˜ andy ˜ lies entirely inside K. In particular, the pointz ˜ lies inside K. Note, that this point belongs also to the hyperplane Ws, whence also to the set Ks. But this means that π(˜s) = (1 − λ)x + λy ∈ π(Ks) and hence the claim follows. Next, since the sets π(Ks), π(Kr) and π(Kt) lie in the same hyperplane, we are allowed to use the additive version of the Brunn–Minkowski inequality (see [Fed69], Theorem 3.2.41):

n−1 1 n−1 1 n−1 1 H (Kt) n−1 = H (π(Kt)) n−1 ≥ H (λπ(Kr) + (1 − λπ(Kt))) n−1 ≥ n−1 1 n−1 1 λ(H (π(Kr))) n−1 + (1 − λ)(H (π(Kt))) n−1 = n−1 1 n−1 1 λ(H (Kr)) n−1 + (1 − λ)(H (Kt)) n−1 , and the lemma follows.

Note that the sets B ∩ (M + g1) and B ∩ (M + g2) lie in the same hyperplane (namely in the hyperplane span{M + g1, g1g2}). We apply Lemma 4.2.1 to this pair of sets and to the space span{M + g1, g1g2}. Hence, taking into account estimate (4.5) we obtain the following chain of inequalities:

n−2 ρ3(φ3) ≥ H (α[B ∩ (M + g1)] + (1 − α)[B ∩ (M + g2)]) ≥

1 1 n−2 αρ1(φ1) n−2 + (1 − α)ρ2(φ2) n−2 .

We continue our estimates now with the help of the Young inequality:

n−2 α 1−α α 1−α ρ3(φ3) ≥ ρ1(φ1) n−2 ρ2(φ2) n−2 = ρ1(φ1) ρ2(φ2) . (4.6)

The inequality that we have just derived is the main part of the proof of the theorem. Note that since the set B is bounded, in the deﬁnition of the integrals Vi one can consider only bounded domains (intervals) of integration (i.e. the intervals, where ρi(φ) 6= 0 or in other words we can consider only those φ for which B ∩ (M + φfi/|fi|) 6= ∅). For i = 1, 2 we reparameterize the functions ρi in a way that both sets over which we integrate in Vi, i = 1, 2 become the interval (0, 1). To this end, for each t ∈ (0, 1) we deﬁne implicit functions φ1 = φ1(t) and φ2 = φ2(t) by the formulae

φ1(t) φ2(t) 1 Z 1 Z t = ρ1(u)du and t = ρ2(u)du. V1 V2 0 0 38 Busemann’s theorem and some of its consequences.

Since ρ1 and ρ2 are non–negative functions, we infer that the functions φ1 and φ2 are strictly monotone (increasing), hence they are also diﬀerentiable for almost all t ∈ R. On R t the other hand, since the functions ρi are continuous, the antiderivatives Ri(t) = 0 ρi(t)dt are diﬀerentiable for all t ∈ (0, 1) and i = 1, 2. Hence for almost all t ∈ (0, 1) one has

 φ1(t)  Z 1 d 1 d 1 dφ1 1 =  ρ1(u)du = (R1(φ1(t))) = ρ1(φ1(t)) . V1 dt V1 dt V1 dt 0

Since the very same estimate holds for the function ρ2, we can calculate the derivatives of the functions φ1 and φ2 for almost all t ∈ (0, 1): dφ V dφ V 1 = 1 and 2 = 2 . (4.7) dt ρ1(φ1) dt ρ2(φ2)

We deﬁne function φ3 with the help of the identity (4.3):

|f3| φ3(t) = . |f1| + |f2| φ1(t) φ2(t)

Next we use (4.7) in order to conclude that φ3 is diﬀerentiable almost everywhere and dφ3 further to calculate the derivative dt for almost all t ∈ (0, 1):

|f3| dφ3 |f1| dφ1 |f2| dφ2 |f1|V1 |f2|V2 2 = 2 + 2 = 2 + 2 . (4.8) φ3 dt φ1 dt φ2 dt φ1ρ1(φ1) φ2ρ2(φ2)

Having obtained the estimates for the derivatives of the functions φi, i = 1, 2, 3, we proceed to the estimate of the term V3/|f3| from the inequality (4.1). We ﬁrst use the deﬁnition of the term V3 and the inequalities (4.6) and (4.8):

1 1 1 Z Z Z V3 1 1 dφ3 1 α 1−α dφ3 = ρ3(t)dt = ρ3(φ3(t)) dt ≥ ρ1(φ1) ρ2(φ2) dt ≥ |f3| |f3| |f3| dt |f3| dt 0 0 0 1 Z 2 2 1 α 1−α |f1|V1φ3 |f2|V2φ3 ρ1(φ1) ρ2(φ2) 2 + 2 dt. |f3| φ1ρ1(φ1)|f3| φ2ρ2(φ2)|f3| 0 Let us now use Young’s inequality and the identity (4.3):

1 V Z ρ (φ )αρ (φ )1−α V φ2 α V φ2 1−α |f | 3 ≥ 1 1 2 2 1 3 2 3 3 dt = |f3| |f3| φ1ρ1(φ1)|f3| φ2ρ2(φ2)|f3| φ3 0 1 Z V α V 1−α φ = 1 2 3 dt. φ1 φ2 |f3| 0 4.3 Area non–increasing projections in codimension one. 39

Thanks to the Young inequality and to the formulae (4.4), we are ready now to ﬁnish oﬀ the proof of the inequality (4.1): 1 V Z αφ (1 − α)φ −1 φ |f | |f |−1 3 ≥ 1 + 2 3 dt = 1 + 2 . |f3| V1 V2 |f3| V1 V2 0

Hence we have just proved that |f1|/V1 + |f2|/V2 ≥ |f3|/V3 for all f1, f2 and f3 such that f3 = f1 +f2. Note that this inequality is not exactly what we need in the formulation of the n−1 ⊥ Busemann theorem, since apriori Vi is not equal to H (B ∩ fi ). We reduce the desired ⊥ inequality to the inequality (4.1). To this end, we deﬁne lines li, i = 1, 2 as li := L ∩ fi and denote the corresponding unit vectors by hi, i = 1, 2. Note that hi is orthogonal to fi. ∗ Next we take the vectors fi := |fi|hi and apply to them the inequality (4.1): |f | |f | |f ∗| |f ∗| |f ∗| |f | 1 + 2 = 1 + 2 ≥ 3 = 3 , V1 V2 Ve1 Ve2 Ve3 V3 n−1 ⊥ where Vei := H (B ∩ fi ). This ﬁnishes the proof of Theorem 4.2.1.

4.3 Area non–increasing projections in codimension one.

Next we present another theorem of Busemann which we will frequently use afterwards. n−1 This result will provide us with a formula for the Hausdorﬀ measure H||...||(U) of the Borel subsets of the hyperplanes. Theorem 4.3.1. (Busemann) Let (X, || ... ||) be an n–dimensional Banach space with the unit ball B = {x ∈ X : ||x|| ≤ 1}, and let W be a (n − 1)–dimensional linear subspace of X. Then, for every Borel subset U ⊂ W the following formula holds true: Hn−1(U) Hn−1(U) = φ (W ) ·Hn−1(U) = α . ||...|| BH n−1 Hn−1(B ∩ W ) Proof. Let us write Hn−1(U) µ (U) := α B,W n−1 Hn−1(B ∩ W ) n−1 for brevity. Note that both µB,W and H||...|| W are Haar measures on W . According to [Mat99] Theorem 3.1 this means that these measures are proportional. Hence it is n−1 suﬃcient to show that µB,W (B ∩ W ) = H||...|| W (B). Since µB,W (B ∩ W ) = αn−1, we are n−1 done once we prove that H||...|| W (B) = αn−1. To this end, we prove one more lemma. Lemma 4.3.1. Let (X, || ... ||) be an n–dimensional Banach space with the unit ball B = {x ∈ X : ||x|| ≤ 1}, and let λ be a Haar measure on X. Then for all measurable sets U ⊂ X one has λ(U) Hn (U) = α . ||...|| n λ(B) 40 Busemann’s theorem and some of its consequences.

Proof. We shall prove two inequalities, namely λ(U) λ(U) Hn (U) ≤ α and Hn (U) ≥ α . ||...|| n λ(B) ||...|| n λ(B) We start with the ﬁrst one. Fix ε > 0 and δ > 0. Note that since U is measurable, there exists an open set G(= Gε) such that U ⊂ Gε and λ(G\U) < ε/2. Hence, according to 0 the Vitali covering theorem, there also exist countable collections of points xi ∈ U and of 0 0 0 0 0 numbers ρi < δ/2 such that the balls B(xi, ρi) = B||...||(xi, ρi) are pairwise disjoint and the S∞ 0 0 set C := i=1 B(xi, ρi) enjoys C ⊂ G and λ(U\C) = 0. Furthermore, since λ(U\C) = 0, 00 00 there exist another countable collections of points xi and of numbers ρi < δ/2 such that S∞ 00 00 U\C ⊂ i=1 B(xi , ρi ) and ∞ X ε λ(B(x00, ρ00)) < . i i 2 i=1

Combining these collections into one family {B(xi, ρi), i = 1, 2,...} we infer that there S∞ holds U ⊂ i=1 B(xi, ρi) and therefore ∞ X λ(B(xi, ρi)) < ε + λ(U). i=1 From here we conclude that ∞ X αnε λ(U) α ρn < + α . n i λ(B) n λ(B) i=1 Since ε > 0 is arbitrary, the ﬁrst inequality follows. Next we proceed to the reciprocal inequality. For its proof we shall need an auxiliary result, which is called the isodiametric inequality. Lemma 4.3.2. (Isodiametric inequality) Let λ be a Haar measure on a d– dimensional Banach space (X, || ... ||) with the unit ball B = {x ∈ X : ||x|| ≤ 1}, and let A be a measurable subset of X. Then

2dλ(A) ≤ (diam(A))dλ(B).

Proof. Firstly, the inequality is trivial if either A is null set, or A is unbounded, hence we may consider only bounded subsets A ⊂ X of positive measure. Secondly, since diam(A) = diam(cl(A)) (where by cl(A) we mean the closure of the set A), it is suﬃcient to prove the inequality for closed and bounded (i.e. compact) sets. Thirdly, since the distance function is continuous, for a compact set A its diameter is attained: there exist some a1 ∈ A and a2 ∈ A such that ρ = diam(A) = ||a1 −a2||. Hence the ball B||...||(a1, ρ) contains A. Since the set B||...||(a1, ρ) is convex, it also contains conv(A), the convex hull of the set A. Since a1 was an arbitrary point, at which the diameter is attained, we conclude that diam conv(A) = diam(A). Since λ(conv(A)) ≥ λ(A), we infer that it is left to prove Lemma 4.3.2 for convex sets. 4.3 Area non–increasing projections in codimension one. 41

Let K be a convex body with the diameter ρ. Hence ||x − y|| ≤ ρ for all x, y ∈ K. Therefore, K + (−K) ⊂ ρB, and the Brunn–Minkowski inequality gives

d d 1 1 d 2 λ(K) = λ(K) d + λ(−K) d ≤ λ(K + (−K)) ≤ ρ λ(K), which ﬁnishes the proof of Lemma 4.3.2.

Now let Ai be an arbitrary covering of the set U by sets of diameter less than δ. Then ∞ ! ∞ ∞ n [ X X diam(Ai) λ(U) ≤ λ A ≤ λ(A ) ≤ λ(B), i i 2 i=1 i=1 i=1 where the last inequality follows from Lemma 4.3.2. Hence Lemma 4.3.1 follows. In order to finish the proof of Theorem 4.3.1 it suffices to apply Lemma 4.3.1 to the space X defined as X := (W,| ...|), where the norm | ...| is the one given by the convex set B ∩ W , the set U defined as the intersection between B and W and any Haar measure λ on the space X. The following theorem was once again first discovered by Busemann. This result will play crucial role in the following two chapters, especially in the proofs of the existence for the Plateau problem in codimension one. The proof presented here is the one taken from the article [Bus50]. Theorem 4.3.2. (Busemann) Let (X, || ... ||) be an n–dimensional Banach space and let W be a (n − 1)–dimensional linear subspace of X. Then there exists a linear projection n−1 n−1 πW : X → W with the range W, such that H||...||(πW (A)) ≤ H||...||(A) for any Borel subset A ⊂ V, where V is some (n − 1)–dimensional linear subspace of X. Proof. We construct the desired projection by a more or less direct procedure. In more details, the projection will resemble a nearest point projection with respect to some norm given by a very special convex set. We start by constructing this convex set. Note that, thanks to Theorem 4.2.1, we know that the set K = {x ∈ X : σB(x) ≤ 1} is convex. We remind to the reader the definition of the function σB : X → R+, σB(x) := n−1 ⊥ ⊥ αn−1|x|/H (B ∩ x ), where by x we denote the hyperplane orthogonal to the vector x. Since the ball K contains the point 0, we can introduce its polar dual set. In more details, the set K∗ := {x ∈ X : hx, yi ≤ 1 for all y ∈ K} is called the polar dual of K. Note that K∗ is a convex symmetric set, containing the point 0. Therefore, it gives rise to the norm || ...||, simply by setting K∗ to be the unit ball of this norm. Next, given a hyperplane W, let ω be a vector such that W is a supporting hyperplane (see 3.1.2) of the ball B||...||(ω,||ω||) at the point 0. We claim that the projection parallel to the vector ω is area non–increasing. In order to prove this statement, we shall need two more definitions. First we introduce the supporting function S(K, ·): X → R+ of the set K as S(K, u) := suphu, xi. x∈K 42 Busemann’s theorem and some of its consequences.

Notice that the value S(K, u) is equal to the distance from the point 0 to a supporting hyperplane H of the set K at some point x0 ∈ K such that the unit outward normal vector to H is equal to u/|u|. Let us also deﬁne function ρ(K, ·): X → R+, which is called Minkowski functional of K, as

ρ(K, u) := sup{λ ∈ R+ : λu ∈ K}.

We infer that since K and K∗ are polar dual, one has S(K∗, u) = 1/ρ(K, u). Hence, taking into account the deﬁnition of the set K, we conclude that

1 = ρ(K, u) = Hn−1(B ∩ u⊥). (4.9) S(K∗, u)

Note that it is left to prove the following statement: if Z is a cylinder deﬁned as Z = π−1(A) = {x ∈ X : x = a + tω, a ∈ A, t ∈ R}, then

n−1 n−1 H||...||(Z ∩ V ) ≥ H||...||(A), (4.10) where V and A are as in the formulation of the theorem. We rewrite both sides of (4.10) using Theorem 4.3.1:

n−1 n−1 ⊥ n−1 H (Z ∩ V ) H (Z ∩ ω ) H||...||(Z ∩ V ) = n−1 = n−1 , H (B ∩ V ) H (B ∩ V ) cos(ω, V⊥) n−1 n−1 ⊥ n−1 H (A) H (Z ∩ ω ) H||...||(A) = n−1 = n−1 , H (B ∩ W ) H (B ∩ W ) cos(ω, W⊥) where by V⊥ we denote the outward unit normal vector of the hyperplane V. We collect most of the introduced notations in the picture ?? page 43. Comparing these identities ⊥ with (4.10) and taking into account formula (4.9) applied to u = V⊥ and u = V , we observe that it is left to prove the following inequality

S(K∗,V ) S(K∗,W ) ⊥ ≥ ⊥ . cos(ω, V⊥) cos(ω, W⊥)

But this is an obvious consequence of the deﬁnition of the vector ω. Indeed, the expres- ∗ sion S(K ,V⊥)/ cos(ω, V⊥) is equal to the length of the segment which the hyperplane V ∗ intercepts on the ray r := {tω, t ≥ 0}, and, analogously, S(K ,W⊥)/ cos(ω, W⊥) equals the length of the segment which the hyperplane W intercepts on the same ray r. The endpoint of the ﬁrst segment is outside of the set K∗, whereas the endpoint of the second one belongs to the boundary of K∗.

Remark 4.3.1. Notice that the result of Theorem 4.3.1 is non–trivial. For instance, this can be illustrated by the fact that the Euclidean orthogonal projections machinery seems to be unusefull here. 4.3 Area non–increasing projections in codimension one. 43 44 Busemann’s theorem and some of its consequences.

We prove one more theorem, which is a generalization of the formula from the formulation of Theorem 4.3.1 to the case of the (n − 1)–rectiﬁable subsets of X. We shall need this result while proving Theorem 4.3.4. This theorem was ﬁrst formulated and proved in the present thesis by I.Vasilyev.

Theorem 4.3.3. (I.Vasilyev) Let (X, || ... ||) be an n–dimensional Banach space. If A is a (n − 1)–rectiﬁable subset of X, then the following formula holds true: Z n−1 n−1 H||...||(A) = φBH (TxA)dH (x). A

Remark 4.3.2. Note that the integrand here has sense since TxA ∈ G(n − 1,X) exists almost everywhere, thanks to Theorem 3.2.6.

Remark 4.3.3. Theorem 4.3.2 holds in a more general setting. Namely it is also true for m–rectiﬁable subsets of X for all 1 ≤ m ≤ n. Since we are not using this generalization in this thesis, we do not prove it here.

Proof. We need one technical lemma before starting the proof of the theorem.

1 Lemma 4.3.3. Let M ⊂ X be a (n − 1)–dimensional C –manifold, and let x0 ∈ M. Then

n−1 H||...||(M ∩ B(x0, r)) φBH (Tx0 M) = lim n−1 . r→0 αn−1r

Proof. With no loss of generality we suppose x0 = 0. For sake of brevity we write H := T0M. We also ﬁx an orthonormal basis e1, . . . , en of the space X, in a way that e1, . . . , en−1 becomes a basis of H. First of all note that since M is a C1–manifold of dimension n − 1, it is locally a Lipschitz graph. In more details, for every ε > 0 there exists r0 = r0(ε) such ⊥ that for every r < r0 the set (M ∩ (H ∩ B(0, r)) + H ) can be written as the image of the ⊥ map F : H ∩ B(0, r) → M ∩ ((H ∩ B(0, r)) + H ) given by the formula F (x) = x + enf(x) for some function f : H ∩ B(0, r) → R, which is ε–Lipschitz. Second of all we claim that for every ε > 0 and for every r < r0(ε) there holds

n−1 n−1 n−1 (1 + ||en||ε) H||...||(H ∩ B(0, r)) ≥ H||...||(F (H ∩ B(0, r))) ≥ n−1 n−1 (1 − ||en||ε) H||...||(H ∩ B(0, r)). (4.11)

Indeed, note that for all x, y ∈ H ∩ B(0, r) one has

||x − y|| + |f(x) − f(y)| · ||en|| ≥ ||F (x) − F (y)|| ≥ ||x − y|| − |f(x) − f(y)| · ||en||.

But the function f(x) is ε–Lipschitz on H ∩ B(0, r). As a consequence of this fact we get that (1 +ε ˜) ||x − y|| ≥ ||F (x) − F (y)|| ≥ (1 − ε˜) ||x − y||, 4.3 Area non–increasing projections in codimension one. 45

whereε ˜ = ||en||ε and our claim (4.11) follows. Third of all, we infer that for the ﬁxed ε there exists r1 = r1(ε) such that for all r < r1(ε) one has

n−1 n−1 n−1 H||...||(M ∩ B(0, r)) ≤ H||...||(F (H ∩ B(0, r))) ≤ ε + H||...||(M ∩ B(0, r)). (4.12) Since the first inequality is trivial, let us concentrate on the second one. Since M ∩ n−1 n−1 B(0, r) ⊆ F (H ∩ B(0, r)), and since the measures H and H||...|| are proportional (as n−1 both being Haar invariant measures), it is sufficient to show that the measure H (Er), where Er := F (H ∩ B(0, r))\(M ∩ B(0, r)) is small for r small enough. To this end, we first infer that there exists (1 − ε) ≤ γ ≤ 1 such that for all r < r0 there holds PH (Er) ⊆ B(0, r)\B(0, γr), where PH is the Euclidean projection with the range H. For instance one can take γ = 1/(1 + ε) thanks to the fact that F is (1 + ε)–Lipschitz in the ball B(0, r0). Hence we have that n−1 n−1 n−1 n−1 H (PH (Er)) ≤ αn−1 (γr) − r ≤ Cr ε with a constant C depending on n only. Recall that since F is (1 + ε)–Lipschitz, the n Jacobian J F (see 3.2.5) satisfies J F (x) ≤ (1 + ε) for all x ∈ B(0, r0). Now we estimate, using the area formula: Z n−1 n−1 n−1 H (Er) = J F (x)dH (x) ≤ Cεre ,

PH (Er) where the constant Ce does not depend neither of ε, nor on r. Our claim (4.12) now follows. On the other hand,

n−1 n−1 n−1 H||...||(H ∩ B(0, r)) = φBH (H)H (H ∩ B(0, r)) = φBH (H)αn−1r . (4.13) Our conclusion now follows from the estimates (4.11), (4.12) and (4.13). Now we proceed to the proof of Theorem 4.3.3. We ﬁrst prove the theorem in case when A is piece of a smooth manifold. Lemma 4.3.4. Let M be a (n − 1)–dimensional C1–submanifold of X, and let A ⊆ M be a Borel set. Then the following formula holds true Z n−1 n−1 H||...||(A) = φBH (TxM)dH (x). A Proof. First, in case when Hn−1(A) = ∞ the lemma is trivial. Indeed, this follows from the fact that there exists a constant C > 0 such that for all W ∈ G(n − 1,X) there holds φBH (W ) ≥ C. Indeed this a consequence of the fact that φBH is a continuous function deﬁned on the compact set G(n − 1,X). Next we assume Hn−1(A) < ∞ and further choose an open set U such that A ⊆ U and Hn−1(M ∩ U) < ∞. Fix ε > 0. Thanks to Theorem 4.2.1, we know that the function φBH : G(n − 1,X) → R is convex, hence also continuous. Referring to this fact and also to Lemma 4.3.3, we infer that for every x ∈ A there exists ρx such that 46 Busemann’s theorem and some of its consequences.

1.0 < ρx ≤ ε,

2. M ∩ B(x, ρx) ⊆ U,

3. |φBH (TxM) − φBH (TyM)| ≤ ε for all y ∈ M ∩ B(x, ρx),

4. for all 0 < r ≤ ρx,

n−1 H (M ∩ B(x, r)) ||...|| − φ (T M) ≤ ε, n−1 BH x αn−1r

5. for all 0 < r ≤ ρx, n−1 H (M ∩ B(x, r)) n−1 − 1 ≤ ε, αn−1r where the last condition follows from the fact that M is a smooth manifold. Thus the collection B = {B(x, r): x ∈ A, 0 < r < ρx} is a Vitali covering of the set A. Hence by the Besicovitch theorem 3.2.9 there exists a countable system of disjoint balls {B(xi, ri)}i∈I ∈ n−1 S B such that H (A\ i∈I B(xi, ri)) = 0. Moreover, since φBH is continuous and since the set G(n − 1,X) is compact, we conclude that there exists a constant C0 such that φBH (TxM) ≤ C0 for all x ∈ M. Now we estimate

Z n−1 n−1 H (A) − φBH (TxM)dH (x) = ||...|| A

X X Z Hn−1(A ∩ B(x , r )) − φ (T M)dHn−1(x) ≤ ||...|| i i BH x i∈I i∈I A∩B(xi,ri)  

X Z  φ (T M)α rn−1 − φ (T M)dHn−1(x) + εα rn−1 ≤  BH xi n−1 i BH x n−1 i  i∈I A∩B(xi,ri)   X Z  |φ (T M) − φ (T M)|dHn−1(x) + (1 + φ (T M))εα rn−1 ≤  BH xi BH x BH xi n−1 i  i∈I A∩B(xi,ri) X 1 X ε Hn−1(M ∩ B(x , r )) + (1 + C )ε Hn−1(M ∩ B(x , r )) ≤ i i 0 1 − ε i i i∈I i∈I 1 + C ε 1 + 0 Hn−1(M ∩ U). 1 − ε

Since ε > 0 is arbitrary, the lemma follows. 4.3 Area non–increasing projections in codimension one. 47

Now we derive Theorem 4.3.3 from Lemma 4.3.3. Note that the rectifiability theorem 1 n−1 S tells us that there exist countably many C –manifolds Mi such that H (A\ i Mi) = 0 and for almost every x ∈ A∩Mi one has TxA = TxMi, see [Fed69], Theorem 3.2.19. Next we S S infer that there exists a disjoint union A = i Ai ∪ (A\ i Ai) such that Ai ⊂ Mi are Borel n−1 S sets and H (A\ i Ai) = 0. Indeed, we first define A1 = A ∩ M1, next A2 = A ∩ M2\A1, then A3 = A ∩ M3\(A1 ∪ A2) and so on and so far. Now it is obvious that Ai are pairwise S S n−1 S disjoint and Borel. Moreover, one has A\ i Ai ⊆ A\ i Mi hence H (A\ i Ai) = 0. Since Ai ⊆ Mi for each i we can now use the previous lemma: Z Z n−1 X n−1 X n−1 H||...||(A) = φBH (TxMi)dH (x) = φBH (TxA)dH (x) = i i Ai Ai Z n−1 φBH (TxA)dH (x), A and the proof is complete. The notion of good projections that we introduce in the following definition is going to be very important for us. These projections will play crucial role in the next chapter. Definition 4.3.1. (Good projections) Let (X, || ... ||) be an n–dimensional Banach space and let W ∈ G(n − 1,X). We say that a linear projection πW : X → W is W –good if

n−1 n−1 H||...||(πW (A)) ≤ H||...||(A) for any set A ⊂ V, where V is some (n − 1)–dimensional linear subspace of X. Remark 4.3.4. It follows directly from Theorem 4.3.2 that in every finite dimensional Banach space X each W ∈ G(n − 1,X) admits at least one W –good projection. Next we introduce a number of classical definitions regarding measures on finite dimensional Banach spaces. Definition 4.3.2. (Tangent measures) Let µ be a Radon measure on an n–dimensional Banach space X and let a ∈ X. We say that ν is a m– tangent measure of the measure µ at the point a if one has 1 ν = lim Ta,r,∗µ, r→0 rm where we take the limit in the sense of weak convergence of measures and the image measure Ta,r,∗µ is defined as follows: Ta,r,∗µ(A) = µ(a + r · A). We denote the set of all m–tangent measures of µ at the point a as Tanm(µ, a). Definition 4.3.3. (Densities of measures) Let (X, || ... ||) be an n–dimensional Banach space and let 0 ≤ s ≤ n be a real number. Suppose that µ is measure on X. The upper and lower s–densities of µ at x ∈ X are defined by

s,∗ µ(B||...||(x, r)) Θ||...||(µ, x) = lim sup s , r→0 αsr 48 Busemann’s theorem and some of its consequences.

s µ(B||...||(x, r)) Θ∗,||...||(µ, x) = lim inf s . r→0 αsr If they agree the common value

s s s,∗ Θ||...||(µ, x) = Θ∗,||...||(µ, x) = Θ||...||(µ, x) is called the s–density of µ at x. In the following theorem we establish one important property of the good projections, namely that those projectors do not increase the Hausdorff measure of arbitrary (n − 1)– rectifiable subsets of X. We will further need this result in our first proof of the existence of mass minimising rectifiable chains in codimension one. This result was first formulated and proved in the present thesis by I.Vasilyev. Theorem 4.3.4. (I.Vasilyev) Let (X, || ... ||) be an n–dimensional Banach space. If A is a (n − 1)–rectifiable subset of X, and if πW : X → W is a W –good projection then the following inequality holds true:

n−1 n−1 H||...||(πW (A)) ≤ H||...||(A). Proof. We precede the proof of the theorem with two technical lemmata. ∞ Lemma 4.3.5. In the notation used in the theorem, if q is a norm on X and if {νj}j=1 is a sequence of Radon measures on X that converges weakly in the sense of measures to a measure ν then for all x we have

lim νj(Bq(x, r)) = ν(Bq(x, r)), j→+∞ except of a countable number of r (which may depend on x).

Proof. Note that since νj → ν weakly one has (see [Mat99], Theorem 1.24)

1. ν(O) ≤ lim inf νj(O) for all open subsets O ⊂ X, j→+∞

2. ν(K) ≥ lim sup νj(K) for all compact subsets K ⊂ X. j→+∞ We use these inequalities together with the fact that except of a countable number of r there holds ν(∂Bq(x, r)) = 0:

ν(Bq(x, r)) = ν(int(Bq(x, r))) ≤ lim inf νj(Bq(x, r)) ≤ lim sup νj(Bq(x, r)) ≤ ν(Bq(x, r)), j→+∞ j→+∞ where the last inequality follows from the fact that Bq(x, r) is compact since X is ﬁnite dimensional. (Here int(Bq(x, r)) denotes the interior of the closed ball (Bq(x, r)).) Lemma 4.3.6. In the notation used in the theorem, let µ be a Radon measure deﬁned as n−1 n−1 µ = H||...|| A. Then for µ–almost all x ∈ A one has Tann−1(µ, x) = {H||...|| Wx} for some hyperplane Wx ∈ G(n − 1,X). 4.3 Area non–increasing projections in codimension one. 49

Proof. Denote by µx the unique element of the set Tann−1(µ, x), the uniqueness follows from [Mat99], Theorem 15.19. Since A is rectiﬁable, according to Theorem 4.3.3, we infer 1 n−1 n−1 that there exists a function θ ∈ Lloc(H A) such that µ = θ · µ,e where µe = H A. n−1 Note that for µ–almost all x ∈ A there holds Tann−1(µ,e x) = {H Wx}, for some n−1 Wx ∈ G(n−1,X). We claim that for µ–almost all x ∈ A one also has µx = θ(x)H Wx. Indeed, this follows from [Lel08], Proposition 3.12. As a consequence we conclude that for n−1 1 n−1 µ–almost all x, µx = β(x)H||...|| Wx for some function β ∈ Lloc(H A). On the other hand,

n−1 µ (B (x, r)) α rn−1 Θ (µx, x) β(x) = lim x ||...|| · n−1 = ||...|| . r→0 n−1 n−1 n−1 n−1 αn−1r H||...||(Wx ∩ B||...||(x, r)) Θ||...||(H||...|| Wx, 0)

n−1 Thanks to Lemma 4.3.5, we know that Θ||...||(µx, x) = 1 for µ–almost all x ∈ A. Indeed, for µ–almost all x ∈ A

n−1 µx(B||...||(x, ρ)) µ(B||...||(x, rρ)) n−1 Θ||...||(µx, x) = lim n−1 = lim lim n−1 = Θ||...||(µ, x) = 1, ρ→0 αn−1ρ ρ→0 r→0 αn−1(rρ) where the last identity follows from B.Kirchheim’s theorem 3.2.3. Referring once again to n−1 n−1 B.Kirchheim’s theorem we infer that Θ||...||(H||...|| Wx, 0) = 1 for µ–almost all x, and hence β(x) = 1 for µ–almost all x and the lemma follows.

We proceed to the proof of the theorem keeping the notations µ and µx introduced in Lemma 4.3.6. First, note that we can replace A by a Borelian subset Ae ⊆ A such that for n−1 all x ∈ Ae one has Tann−1(µ, x) = {H Wx},Wx ∈ G(n−1,X). Denote E = πW (A). We n−1 assume, as we can, that H||...||(A) < ∞. Hence, thanks to the Eilenberg inequality (see 3.2.4 n−1 −1 and [Fed69], 2.10.25), for H –almost all y ∈ E there holds card(A ∩ πW {y}) < ∞. We infer that

n−1 1. we can replace E by a set Ee ⊂ E that satisﬁes H||...||(E\Ee) = 0 and such that −1 card(A ∩ πW {y}) < ∞ for all y ∈ E,e

−1 2. we can replace A by A ∩ πW (Ee),

n−1 n−1 3. we can assume that Θ||...||(H||...|| A, x) = 1 for all x ∈ A.

n−1 The ﬁrst replacement changes nothing: since πW is a Lipschitz mapping, it maps H – negligible sets to Hn−1–negligible sets. The second replacement changes nothing as well, n−1 −1 n−1 since H||...||(A ∩ πW (Ee)) ≤ H||...||(A). We can make the third assumption, because we can n−1 n−1 n−1 replace A by a set A1 ⊆ A, satisfying H||...||(A\A1) = 0 and Θ (H||...|| A, x) = 1 for all x ∈ A1, see Lemma 4.3.6. Let us deﬁne a norm q on the space X as q(u) = max{||πW (u)||, ||u − πW (u)||}. Note −1 that for all y ∈ Ee there exists δ = δ(y) > 0 such that for all x1, x2 ∈ A ∩ πW {y} and for 50 Busemann’s theorem and some of its consequences.

all r ∈ (0, δ(y)) one has Bq(x1, r) ∩ Bq(x2, r) = ∅ once x1 and x2 are distinct. Indeed, since −1 A ∩ πW {y} is a ﬁnite set, we can take 1 δ(y) = min q(xl − xm). −1 3 xl6=xm∈A∩πW {y}

Now we claim that for all ε > 0 and for all y ∈ Ee there exists δe = δe(y) > 0 such that for −1 all x ∈ A ∩ πW {y} and for all but countably many r ∈ (0, δe) one has

n−1 |µ (Bq(x, r)) − µx (Bq(0, r))| ≤ εαn−1r . (4.14)

Note that, according to Lemma 4.3.5 for every εe > 0, there exists δe1 = δe1(x) such that for all ρ < δe1 and for all but countably many r ∈ R+ there holds

µ(Bq(x, rρ)) µx (Bq(0, r)) − ≤ ε. (4.15) ρn−1 e

n−1 n−1 On the other hand, since A is rectiﬁable and since Θ||...||(H||...|| A, x) = 1, referring to B.Kirchheim’s theorem 3.2.3, we infer that for the ﬁxed ε > 0 there exists δe2 = δe2(x) such that for all re ∈ (0, δe2) there holds

µ(Bq(x, re)) n−1 − 1 ≤ ε. αn−1re

n−1 Next for r < δe2 we estimate choosing εe = εαn−1r in (4.15) and further taking ρ < min{δe1, δe2/r}:

|µ (Bq(x, r)) − µx (Bq(0, r))| ≤

µ(Bq(x, rρ)) µ(Bq(x, rρ)) n−1 n−1 µx (Bq(0, r)) − + − αn−1r + αn−1r − µ(Bq(x, r)) ≤ ρn−1 ρn−1 n−1 3εαn−1r , and it suffices to take δe(y) = minx∈π−1(y){δe2(x)} in order to finish the proof of the claim (4.14). n−1 n−1 Next, referring to the fact that A is rectifiable and since Θ||...||(H||...|| A, x) = 1, we −1 0 infer that for every y ∈ Ee and for every x ∈ πW (y) for every ε > 0 there exists δ such that for all r < δ0 there holds

n−1 H (A ∩ B(x, r)) ||...|| − φ (W ) ≤ ε. n−1 BH x αn−1r Hence if ε is small enough, for all r ≤ δ0 we can guarantee

n−1 n−1 C0r ≤ H||...||(A ∩ Bq(x, r)), (4.16) 4.3 Area non–increasing projections in codimension one. 51

where the constant C0 depends only on the dimension n. For every y ∈ Ee denote ∆(y) = min{δ(y), δe(y), δ0(y)}. According to the Vitali covering theorem there exists a sequence of balls Bj = Bq(yj, rj) ∩ W in W such that for all j the radii rj satisfy 0 < rj < ∆(yj) and are outside of the bad set of radii from the line (4.14), n−1 S∞ the balls Bj are mutually disjoint and H||...||(Ee\ j=1 Bj) = 0. Hence

∞ n−1 n−1 X n−1 H||...||(πW (A)) = H||...||(Ee) ≤ H||...||(Bj). (4.17) j=1

−1 For each j denote kj = card(A ∩ πW {yj}). Let xj,1, . . . , xj,kj be the elements of the set −1 Skj A ∩ πW {yj}. We write Wj = k=1 xj,k + Wxj ,k for sake of brevity. We claim that

Hn−1(A ∩ B (x , r )) − Hn−1((x + W ) ∩ B (x , r )) ≤ ||...|| q j,k j ||...|| j,k xj,k q j,k j n−1 C1εH||...||(A ∩ Bq(xj,k, rj)), (4.18) for some universal constant C1. Indeed, we ﬁrst note that the inequality (4.14) gives

Hn−1(A ∩ B (x , r )) − Hn−1((x + W ) ∩ B (x , r )) ≤ εα rn−1. ||...|| q j,k j ||...|| j,k xj,k q j,k j n−1 j

0 On the other hand, since rj < δ (y), we conclude from (4.16) that for ε small enough there n−1 n−1 holds εαn−1rj ≤ C1εH||...||(A ∩ Bq(xj,k, rj)) and our claim follows. The line (4.18) now gives:

kj kj X n−1 n−1 [ H||...||(Wj ∩ Bq(xj,k, rj)) = H||...||(Wj ∩ Bq(xj,k, rj)) ≤ k=1 k=1

kj X n−1 (1 + C1ε) H||...||(A ∩ Bq(xj,k, rj)), (4.19) k=1 thanks to the fact that the balls Bq(yj, rj) are mutually disjoint. −1 Let us use the deﬁnition of the norm q. Note that Wj ∩Bq(xj,k, rj) = Wj ∩πW (Bq(yj, rj)) and hence πW (Wj ∩ Bq(xj,k, rj)) = W ∩ Bq(yj, rj). Since the projection πW is W –good we have that n−1 n−1 n−1 H||...||(Bj) = H||...||(W ∩ Bq(yj, rj)) ≤ H||...||(Wj ∩ Bq(xj,k, rj)). This inequality together with the estimate (4.19) allows us to write

kj kj n−1 X n−1 X n−1 H||...||(Bj) ≤ H||...||(Bj) ≤ H||...||(Wj ∩ Bq(xj,k, rj)) ≤ k=1 k=1

kj X n−1 (1 + C1ε) H||...||(A ∩ Bq(xj,k, rj)). k=1 52 Busemann’s theorem and some of its consequences.

We ﬁnish our estimates recalling the line (4.17):

∞ kj n−1 X X n−1 n−1 H||...||(πW (A)) ≤ (1 + C1ε) H||...||(A ∩ Bq(xj,k, rj)) ≤ (1 + C1ε)H||...||(A), j=1 k=1 where the last inequality follows from the fact that the balls {Bq(xj,k, rj): j = 1, 2,... ; k = 1, . . . kj} are mutually disjoint. The theorem will follow after taking a limit when ε tends to 0.

The following theorem, the last one in this chapter, is the one where we derive the fact that good projectors do not increase the Hausdorﬀ mass of (n − 1) polyhedral G chains in X. We will further rely on this result in our proof of the “triangle inequality” for (n − 1) cycles.

Theorem 4.3.5. (I.Vasilyev) Let (X, || ... ||) be an n–dimensional Banach space. If P ∈ Pn−1(X,G) (i.e. P is a (n − 1) polyhedral G chain) and if πW : X → W is a W –good projection then the following inequality holds true:

M(πW #(P )) ≤ M(P ).

Remark 4.3.5. This result was ﬁrst formulated and proved in the present thesis by I.Vasilyev.

Proof. We write π instead of πW here for sake of brevity. Suppose that

N X P = gi[σi] i=1 with σi non–overlapping simplexes. Consider a family of simplexes π(σ1), . . . , π(σN ) and choose a non–overlapping representation of π#P , say

M X π#P = hk[τk]. k=1

−1 Deﬁne for each k from 1 to M a number ak as ak := card(i : π (τk) ∩ int(σi) 6= ∅) and an increasing function φk : [1, . . . , ak] → [1,...,N] deﬁned inductively as follows, −1 φk(1) = min{i : π (τk) ∩ int(σi) 6= ∅} and for all j from 1 to ak

−1 φk(j) = min{i : π (τk) ∩ int(σi) 6= ∅, i > φk(j − 1)}.

−1 ak Hence τk = π(σφk(j) ∩ π (τk)) for each j ∈ [1, . . . , ak]. Deﬁne further m(k) ∈ {φk(j)}j=1 such that n−1 −1 n−1 −1 H||...||(σm(k) ∩ π (τk)) ≤ H||...||(σl ∩ π (τk)) 4.3 Area non–increasing projections in codimension one. 53

ak P for all l ∈ {φk(j)}j=1. Next we estimate using the fact that hk = j gφk(j), the triangle inequality and the fact that π is W –good

M M a X X Xk (π P ) = |h |Hn−1(τ ) ≤ g Hn−1(π(σ ∩ π−1(τ ))) ≤ M # k ||...|| k φk(j) ||...|| m(k) k k=1 k=1 j=1

M ak N M X X n−1 −1 X X n−1 −1 |gφk(j)|H||...||(π(σφk(j) ∩ π (τk))) ≤ |gi| H||...||(σi ∩ π (τk)) ≤ k=1 j=1 i=1 k=1 N X n−1 |gi|H||...||(σi) = M(P ). i=1 54 Busemann’s theorem and some of its consequences. Chapter 5

The “classical” proof of the existence of mass minimizing (n − 1)–rectiﬁable G chains.

This chapter is devoted to the first proof of the existence of mass minimizing (n − 1)– rectifiable G chains in n–dimensional Banach spaces. Remark 5.0.6. The theorems proved in this chapter were first formulated and proved here by I.Vasilyev.

5.1 The lower semicontinuity on the space of (n − 1) polyhedral G chains.

Next we establish a “triangle inequality” for the (n − 1) polyhedral cycles. Theorem 5.1.1. Let (X, || ... ||) be an n–dimensional Banach space with the unit ball B = {x ∈ X : ||x|| ≤ 1} and let G be a complete normed Abelian group. Let P ∈ Pn−1(X,G) be a (n − 1)–cycle (i.e. ∂P = 0) such that

N X P = gj[σj], j=1 where σj are non–overlapping (n − 1)–simplexes. Then N n−1 X n−1 |g1|H||...||(σ1) ≤ |gj|H||...||(σj). j=2

Proof. For j between 1 and N let Wj be the hyperplane such that σj ⊆ Wj and let π : X → W1 be any W1–good projection. Denote X Q = gj[σj]. j≥2 The “classical” proof of the existence of mass minimizing 56 (n − 1)–rectiﬁable G chains.

Hence P = g1[σ1]+Q and further π#P = g1[σ1]+π#Q. Note that π#P is a top–dimensional G chain (see Deﬁnition 3.1.13) in the space (W1,| ...|), where the norm | ...| is deﬁned by the convex set B ∩ W1. But ∂π#P = 0, and by the constancy theorem (see Theorem 3.2.2) we conclude that g1[σ1] = −π#Q. Now we estimate

n−1 X n−1 M(g1)[σ1] = |g1|H||...||(σ1) = M(π#Q) ≤ M(Q) = |gj|H||...||(σj), (5.1) j≥2 where the inequality in (5.1) follows from the fact that π is W1–good, see Theorem 4.3.5. Hence the theorem follows. Next we are going to prove that the “triangle inequality”, obtained in the previous theorem is a necessary and suﬃcient condition for the lower semicontinuity of the mass on the space of polyhedral chains. Theorem 5.1.2. Let (X, || ... ||) be an n–dimensional Banach space, let G be a complete normed Abelian group and let m be an integer such that 1 ≤ m ≤ n−1. Then the following conditions are equivalent:

1. M is lower semicontinuous with respect to ﬂat convergence on Pm(X,G). PN 2. If P = j=1 gj[σj] ∈ Pm(X,G), where σj are non–overlapping, is a cycle (which means that ∂P = 0) then

N m X m |g1|H||...||(σ1) ≤ |gi|H||...||(σj). j=2

Proof. 2 ⇒ 1. We start with a very special case when P = g[σ], i.e. when the chain P consists of only one simplex. Suppose that F(P −Pi) → 0 as i tends to ∞ for some Pi ∈ Pm(X,G). By the ∞ deﬁnition of ﬂat convergence there exists a sequence of m–chains {Qi}i=1 and a sequence ∞ of (m + 1)–chains {Ri}i=1 such that P − Pi = Qi + ∂Ri and M(Qi) + M(Ri) → 0 when i tends to ∞. Note that ∂(P − Pi − Qi) = ∂(∂Ri) = 0, so we can apply 2 to the chain (P − Pi − Qi) and obtain

(P ) ≤ lim inf (Pi + Qi) ≤ lim inf (Pi) + (Qi) ≤ lim inf (Pi), M i M i M M i M and hence 1 follows. We pass to the general case. Suppose that Pi → P in flat norm for a sequence Pi ∈ Pm(X,G). For every j between 1 and N denote by Wj the m–plane containing the simplex σj. Let Fj,η be a (m + 1)–convex polyhedron defined in the following way. If σj = conv{a1,j, . . . , am+1,j}, then we define Fj,η as Fj,η = conv{a1,j, . . . , am+1,j, vn,j} where

m+1 1 X v = a + ηω , n,j n k,i j k=1 5.1 The lower semicontinuity on the space of (n − 1) polyhedral G chains. 57

for some (fixed) nonzero vector ωj orthogonal to the m–plane Wj and a small number η to be defined in a moment. In more details, we claim that there exists η depending on σ1, . . . , σn only such that Fi,η and Fj,η do not overlap once i 6= j. First note that it is sufficient to find for each pair (i, j) a number ηi,j such that Fi,ηi,j and Fj,ηi,j are non– overlapping. Second, consider two simplexes, say σ1 and σ2 (which are non-overlapping by our assumption). Assume, as we can, that σ1 ∩ σ2 6= ∅, hence, possibly after renumbering, there exists 0 < l < m + 1 such that a1,1 = a1,2 = b1, a2,1 = a2,2 = b2, . . . , al,1 = al,2 = bl. Translate both simplexes to zero so that b1 = 0. Next, we infer that there exists a hyperplane W with the unit normal ω such that conv{0, b1, . . . , bl} ⊂ W and hx, ωi > 0 for all x ∈ σ1\conv{0, b1, . . . , bl} and hx, ωi < 0 for all x ∈ σ2\conv{0, b1, . . . , bl}. Note that we are done once we show that there exists η > 0 such that for all x ∈ int(F1,η) there holds hx, ωi > 0 and for all x ∈ int(F2,η) one has hx, ωi < 0. Fix η > 0 and suppose that there exists x ∈ W ∩ int(F1,η). Then

m+1 m+1 ! X 1 X x = α a + α a + ηω k 1,k 0 n 1,k 1 k=1 k=1

with αi0 > 0 for some i0 ∈ {l + 1, . . . , m + 1}, α0 > 0 and hx, ωi = 0. Hence

m+1 X α0 α0 0 = hx, ωi = α + ha , ωi + ηhω , ωi ≥ ha , ωi + α ηhω , ωi, k n 1,k 1 n 1,i0 0 1 k=1

since α0 > 0 and since ha1,i0 , ωi > 0, this gives a contradiction for η small enough, and our claim follows. We further infer that, using a construction which is very similar to the one above, one can suppose that the polyhedrons Fj,η are “two–sided”. In more details, let us define polyhedrons Fgj,η = conv{a1,j, . . . , am+1,j, vn,j, −vn,j}. The same argumentation as above shows us that one can find η small enough to guarantee that Fgj,η and Fgi,η have no overlapping once i 6= j. We shall further write Fj instead of Fgj,η. Next, let uj be a function defined as

c c uj(x) = min{dist∞(PWj (x), σj ), dist∞(x, Fj )},

c where PWj is the orthogonal projection onto Wj, S stands for the complement of a set S and dist∞ is deﬁned in the following way:

dist∞(x, S) = inf ||x − z||∞, z∈S where || ... ||∞ is the supremum norm with respect to the ﬁxed Euclidean structure. Note that it follows from the construction of the polyhedrons Fj that the sets {x : uj(x) > κ} are pairwise disjoint whenever κ > 0. We infer that for every j ∈ [1,...,N] and for every ε > 0 there exists δ > 0 such that for all r ∈ (0, δ) one has

M (gj[σj] − gj[σj ∩ {x : uj(x) > r}]) ≤ ε. (5.2) The “classical” proof of the existence of mass minimizing 58 (n − 1)–rectiﬁable G chains.

We claim that for every j ∈ [1,...,N] there exists δe ∈ [δ/2, δ] and a sequence δi ∈ (δ/2, δ) such that δi → δe

F(P {x : uj(x) > δe} − Pi {x : uj(x) > δi}) → 0, (5.3) when i tends to +∞. Indeed, by the deﬁnition of ﬂat norm, we know that for every εe > 0 there exists i = i(εe) such that for each i ≥ i(εe) there are Ri ∈ Pm+1(X,G) and Qi ∈ Pm(X,G) with M(Qi) + M(Ri) ≤ εe and P − Pi = Qi + ∂Ri. Hence for every ρ > 0 we can write

P {x : uj(x) > ρ} − Pi {x : uj(x) > ρ} = (Qi + ∂Ri) {x : uj(x) > ρ} =

Qi {x : uj(x) > ρ} + ∂(Ri {x : uj(x) > ρ})+

((∂Ri) {x : uj(x) > ρ} − ∂(Ri {x : uj(x) > ρ})) .

From here we see that for every i ≥ i(εe) and for every ρ > 0 one has

F(P {x : uj(x) > ρ} − Pi {x : uj(x) > ρ}) ≤ εe+ MhRi, uj, ρi. (5.4) On the other hand, referring to the slicing properties we infer that for every i there exists δi ∈ (δ/2, δ) such that MhRi, uj, δii ≤ 2M(Ri)/δ. Passing to a subsequence if necessary, we ∞ may assume that the sequence {δi}i=1 converges towards some δe. Note that

lim ||P ||({x : δe > uj(x) ≥ δi}) = 0. i→∞

Hence for the ﬁxed εe there exists number M(= M(εe)) such that ||P ||({x : δe > uj(x) ≥ δi}) ≤ εe once i > M. According to the inequality (5.4) applied to ρ = δi, for each i > M there holds

F(P {x : uj(x) > δe} − Pi {x : uj(x) > δi}) ≤

F(P {x : uj(x) > δi} − Pi {x : uj(x) > δi}) + F(P {x : δe > uj(x) ≥ δi}) ≤

2M(Ri) 2eε ε + + ||P ||({x : δe > uj(x) ≥ δi}) ≤ ε + + ε, e δ e δ e and the claim (5.3) follows. We further derive an upper bound on the mass of the chains gj[σj] first using the inequality (5.2) then the first case of the proof and finally the fact that δi ≥ δ/2: δ M(gj[σj]) ≤ ε + M(P {x : uj(x) > δe}) ≤ ε + lim inf M(Pi {x : uj(x) > }). i→+∞ 2 We continue our estimates by taking the sum over all indexes j between 1 and N in the previous line:

N N X X δ M(P ) = M(gj[σj]) ≤ Nε + lim inf M(Pi {x : uj(x) > }) ≤ Nε + lim inf M(Pi), i→+∞ 2 i→+∞ j=1 j=1 5.1 The lower semicontinuity on the space of (n − 1) polyhedral G chains. 59

where the last inequality follows from the fact that the sets {x : uj(x) > δ/2} are pairwise disjoint. The ﬁrst implication will now follow after we let ε tend to 0. 1 ⇒ 2. We argue by a contradiction. Suppose that there exist g[σ], g1[σ1], g2[σ2], . . . , gN [σn] such that N X M(gi[σi]) < M(g[σ]) (5.5) i=1 and ∂T = 0 where N X T = gi[σi] − g[σ]. i=1

Write [σ] = [x0, . . . , xm]. For every positive natural number n we introduce the following set of multiindexes:

m In = {k = (k1, . . . , km) ∈ N : k1 + ... + km ≤ n − 1}.

Further, for each k ∈ In we call fn,k the map

m X x − x0 x − x0 f (x) := x + k + . n,k 0 i n n i=1

Note that fn,k(σ) ⊆ σ and that Lip(fn,k) = 1/n for each multiindex k. Next, we deﬁne polyhedral chains Sn by the following formula X Sn := g[σ] + fn,k#T.

k∈In

Let us prove that Sn converge to g[σ] in ﬂat norm. First note that since ∂T = 0, referring to the cone construction, we infer that there exists U ∈ Pm+1(X,G) such that ∂U = T. Now we estimate

X X m+1 (U) F(S − g[σ]) = F(∂ f U) ≤ (f U) ≤ (Lip(f )) nm (U) = M , n n,k# M n,k# n,k M n k∈In k∈In and the convergence follows. Since M is lower semicontinuous, we conclude that

lim sup M(Sn) ≤ M(g[σ]). (5.6) n→∞ On the other hand,

N ! ! X X M(Sn) = M(g[σ]) + M fn,k#gi[σi] − M(fn,k#g[σ]) ≤ k∈In i=1 N ! card(In) X (g[σ]) + (g [σ ]) − (g[σ]) . (5.7) M nm M i i M i=1 The “classical” proof of the existence of mass minimizing 60 (n − 1)–rectiﬁable G chains.

Note that card(I ) 1 n → as n → ∞. nm 2

Hence the lines (5.5) and (5.7) now give lim supn M(Sn) < M(g[σ]), which contradicts (5.6).

Corollary 5.1.1. Let (X, || ... ||) be an n–dimensional Banach space and let G be a complete normed Abelian group. Then the mass M is lower semicontinuous with respect to ﬂat convergence on the space Pn−1(X,G). Proof. It suﬃces to recall Theorems 5.1.1 and 5.1.2.

Remark 5.1.1. We pay the reader’s attention to the fact that the present proof of Corol- lary 5.2.1 is not yet available in other codimensions since Theorem 5.1.1 is known only in codimension one.

5.2 The lower semicontinuity on the space of (n − 1)– rectiﬁable G chains.

The following theorem is the main result of this section and of the whole chapter.

Theorem 5.2.1. Let (X, || ... ||) be an n–dimensional Banach space and let G be a complete normed Abelian group. Then the mass M is lower semicontinuous with respect to ﬂat convergence on Rn−1(X,G). Proof. We start with the ﬁrst step where we prove the theorem in case when the limit chain P is polyhedral.

Lemma 5.2.1. Let X and G be the same as in the theorem. If Ti ∈ Rn−1(X,G),P ∈ Pn−1(X,G), and limi→∞ F(Ti − P ) = 0, then

1 M(P ) ≤ lim inf M(Pi) ≤ lim inf M(Ti) + = lim inf M(Ti), i→+∞ i→+∞ i i→+∞ and the lemma follows. The second step is the following version of the strong approximation theorem of Federer (see [Fed69], Theorem 4.2.20). 5.2 The lower semicontinuity on the space of (n − 1)–rectiﬁable G chains. 61

Theorem 5.2.2. Let (X, || ... ||) be an n–dimensional Banach space and let G be a complete normed Abelian group. If T ∈ Rm(X,G) with 1 ≤ m ≤ n then for every ε > 0 there 1 exists a polyhedral chain P ∈ Pm(X,G) and a C –smooth diﬀeomorphism F : X → X such that

• ||F (x) − x|| ≤ ε for all x ∈ X and F (x) = x once dist(x, set(T )) ≥ ε;

• max{Lip(F ), Lip(F −1)} ≤ 1+ε, where the Lipschitz constants are taken with respect to the norm || ... ||;

• M(F#T − P ) ≤ ε. Proof. We ﬁrst prove a lemma where we construct a very special parametrisation of pieces of smooth manifolds. Lemma 5.2.2. Let X be as in the formulation of the theorem, let M ⊂ X be a C1–smooth 1 manifold and let a ∈ M. Fix t ∈ (0, 1). There exists r0 and a C –smooth diﬀeomorphism F : X → X such that for all r ∈ (0, r0) there holds

• for all x 6∈ U(a, r) one has F (x) = x; • max{Lip(F ), Lip(F −1)} ≤ 1/t where both Lipschitz constants are taken with respect to the norm || ... ||;

• U(a, tr) ∩ M = U(a, tr) ∩ F −1(a + Tan(M, a)).

We remind to the reader that by U(a, r) we mean an open Euclidean ball centred at a with radius r.

Proof. Throughout this proof we write Cn for a constant depending on n only. We present here a (little bit modiﬁed to our setting) proof, borrowed from H.Federer’s book (section 3.1.23). Suppose, as we can, that a = 0 and denote W = Tan(M, 0). Let P : W → X be the orthogonal projection with the range W (we mean the orthogonal projection with respect to the ﬁxed Euclidean structure). Note that for every ε there exists ∗ r1 such that for all ξ ∈ U(0, r1) there holds |||Dψ(ξ) − p ||| ≤ ε, where we denote

−1 ψ := (P (M ∩ U(0, r1))) : P (M ∩ U(0, r1)) → M ∩ U(0, r1) and p∗ := Dψ(0) : W → X for brevity. We remind to the reader that we fix the symbol for restrictions of applications and that the operator norm ||| ... ||| of a linear mapping L : X → X is defined here as |||L||| := sup{||L(h)|| : ||h|| ≤ 1}. Define for r ≤ r1 a function u : X → [0, 1] as |x| u(x) = γ , r ∞ 1 where γ is any C smooth function satisfying γ(z) = 1 for z ∈ [0, t], γ(1) = 0, Lip(γ) ≤ 1−t and γ is strictly decreasing on [t, 1]. The “classical” proof of the existence of mass minimizing 62 (n − 1)–rectifiable G chains.

Choose ε = ε(t) := (1 − t)(1/t − 1)/(Cn(2 − t)) and put r0 = r0(t) = r1(ε(t)). We are now ready to define the desired function F . Let F (x) = x for x∈ / U(0, r) and let F (x) = x+u(x)·[p∗(P (x))−ψ(P (x))] otherwise. Let us check the assertions of the lemma. The first one follows easily from the construction of F ; the third one follows from the facts that u(x) = γ (|x|/r) = 1 for |x| ≤ tr ≤ r and that ψ(P (x)) = x once x ∈ M ∩ U(0, r) and since the image of p∗ is contained in W . Hence it is left to estimate the Lispchitz constants of F and F −1. To this end, we first calculate the differential of F :

∗ ∗ DF (x) = idX + Du(x)[p (P (x)) − ψ(P (x))] + u(x)[p − Dψ(P (x))](P (x)).

Next we estimate the norm of the diﬀerence between DF and idX :

∗ ∗ |||DF (x) − idX ||| ≤ |||Du(x)(p (P (x)) − ψ(P (x))) ||| + |||u(x)(p − Dψ(P (x))) ◦ P ||| =:

|||L1||| + |||L2|||.

We estimate the operators L1 and L2 separately. We ﬁrst treat L1. If h ∈ X such that ||h|| ≤ 1 then

Lip(γ)|h| εC |x| ||L (h)|| ≤ |Du(x)(h)| · ||p∗(P (x))) − ψ(P (x))|| ≤ ε||P (x)|| ≤ n , 1 r (1 − t)r where the last inequality follows from the fact that P is orthogonal and since Lip(γ) = 1/(1 − t). Next we proceed to the term L2:

∗ ∗ |||L2||| = ||u||∞ · |||(p − Dψ(P (x))) ◦ P ||| ≤ |||P ||| · |||p − Dψ||| ≤ Cnε.

As a consequence we conclude that for all x ∈ B(0, r) there holds (1 − t) 1 Cn|x| 1 |||DF (x) − idX ||| ≤ − 1 Cn + ≤ − 1 . Cn(2 − t) t (1 − t)r t

We introduce one more abbreviation, for s ∈ (0, 1) and x, y ∈ X we write

b(s) = F (x + s(y − x)) − s(y − x).

0 Hence it follows that b (s) = DF (x + s(y − x))(y − x) − idX (y − x). Now we estimate for x, y ∈ B(0, r):

1 Z ||F (x) − F (y)|| − ||x − y|| ≤ ||F (x) − x|| + ||F (y) − y|| = || b0 (s)ds|| =

0 1 Z 1 || (DF (x + s(y − x)) − id )(y − x)ds|| ≤ − 1 ||y − x||, X t 0 5.2 The lower semicontinuity on the space of (n − 1)–rectiﬁable G chains. 63

where the last inequality holds true since x + s(y − x) = ys + x(1 − s) ∈ B(0, r0). Hence 1 ||F (x) − F (y)|| ≤ ||y − x||. t On the other hand, the last line implies that ||F (x + v) − F (x) − v|| ≤ (1/t − 1) ||v||. This estimate combined with the fact that ||F (x + v) − F (x) − v|| ≥ ||v|| − ||F (x + v) − F (x)|| implies ||F (x + v) − F (x)|| ≥ 1/t, which ﬁnishes the proof of the lemma.

Let us now derive the approximation theorem from Lemma 5.2.2. Since T ∈ Rm(X,G), we know that T = θ(x)Hm M, for some m–rectiﬁable set M which we will call set(T ) and a Borel measurable function θ : set(T ) → G such that θ ∈ L1(X,G) (which means that the function |θ| ∈ L1(Hm set(T ))). Since set(T ) is m–rectiﬁable, there exist countably many 1 ∞ m C –smooth manifolds {Mi}i=1 and a set M0 such that H (M0) = 0 satisfying [ set(T ) ⊆ Mi ∪ M0. i≥1

Let us next consider a subset Mf ⊆ set(T ) such that for all a ∈ Mf one has Θm(Hm M, a) = m m ∞ Θ (H set(T ), a) = 1 for some M = M(a) ∈ {Mi}i=1. Note that we can guarantee Hm(set(T )\Mf) = 0. Next, for each a ∈ Mf we define r = r(a) such that for all r ∈ (0, r(a)) one has ||T || (B(x, r)\(M(a) ∩ B(a, r))) ≤ ε||T || (B(a, r)) , (5.8) where by ||T || we, as always, denote the measure ||T || = |θ|Hm set(T ). Let us next consider the following family of balls: B = {B(a, r): a ∈ M,f 0 < r < r(a)}. Note that B is a Vitali covering of the set M.f According to the Vitali theorem, we can find a disjoint countable covering, say {B(aj, rj): aj ∈ M,f 0 < rj < r(aj)}. We further renumber the ∞ family of manifolds {Mi}i=1 in a way that for each j ≥ 1 the point aj belongs to the manifold Mj. Let us now use the lemma: for each j ≥ 1, for each triplet (aj,Mj,B(aj, rj)) 1 and for each t ∈ (0, 1) there exists a C –diffeomorphism Fj : X → X satisfying the claims S m of Lemma 5.2.2. Since set(T ) = j (Mj ∩ B(aj, rj)) ∪ N with H (N) = 0, we can find a number N = N(ε) such that ∞ ! X M Tj ≤ ε, (5.9) j=N+1 where we denote Tj = T (Mj ∩ B(aj, rj)) for sake of brevity. The next step consists in proving that for every δ > 0 there exists a polyhedral chain Ψj ∈ Pm(X,G) such that M(Fj#Tj − Ψj) ≤ δ. (5.10) −1 m First of all, we know that ||Fj#Tj|| = (Fj ◦ θ)H Fj(Mj ∩ B(aj, rj)). Since the function θ is integrable on the set Mj ∩ B(aj, rj), according to the Lusin theorem (contact [Fed69] Theorem 2.3.5) for each δ > 0 we can find a compact subset C ⊂ Fj(Mj ∩ B(aj, rj)) such −1 that (Fj ◦ θ) C is continuous and that ||Fj#Tj||(Fj(Mj ∩ B(aj, rj))\C) ≤ δ. Suppose, The “classical” proof of the existence of mass minimizing 64 (n − 1)–rectifiable G chains. as we can that C ⊆ {x : θ(x) 6= 0}. Second of all, we decompose C into a finite union of sets in a way that K [ C = Ci ∪ C0 i=1 with ||Fj#Tj||(C0) ≤ δ and osc(θ Ci) ≤ δ infC |θ|. Third of all, for each natural i between 1 and K there exists Si ⊂ Fj(Mj ∩ B(aj, rj)), a finite union of dyadic m–semi–cubes such m m that H (Ci4Si) ≤ δH (Ci). Finally, for each i ∈ [1,...,K] we choose an element θi ∈ G that guarantees |θi − θ(x)| ≤ δ infCi |θ| for each x ∈ Ci. We define the desired polyhedral chain as K X Ψj = θi[Si]. i=1 Let us estimate the mass of the following difference

Z K m X m M(Fj#Tj − Ψj) ≤ |θ(x)|dH (x) − |θi|H (Si) ≤ i=1 Fj (Mj ∩B(aj ,rj )) K Z K X m X m δ + |θ(x)|dH (x) − |θi|H (Si) ≤ i=1 i=1 Ci K X m m δ + δ||Fj#Tj||(C) + |θi| (H (Ci) − H (Si)) ≤ δ + δ(2 + δ)||Fj#Tj||(C), i=1 and the claim (5.10) follows. Now we are ready to ﬁnish the proof of the theorem. We choose

N X F := F1 ◦ ... ◦ FN and P := Ψj. j=1 By the construction we see that F and P satisfy the ﬁrst two claims of the theorem. Let us check the third one:

∞ N X X M(F#T − P ) = M(F# T B(aj, rj) − P ) ≤ M( Fj#T B(aj, rj) − P ) + ε ≤ ..., j=1 j=1 where the last inequality follows from the line (5.9). We continue the estimate, ﬁrst using the estimate (5.8) and afterwards the inequality (5.10) with δ = ε/N :

N N ! N X X X Nε ... ≤ F T − Ψ + ε ||T || (B(a , r )) + ε ≤ + ε + ε (T ). M j# j j j j N M j=1 j=1 j=1 Since ε > 0 is arbitrary, the theorem follows. 5.3 Solution to the Plateau problem whose support stays inside of the convex hull of the boundary. 65

The third step of the proof of Theorem 5.2.1 consists in derivation of this result from Theorem 5.2.2. Let Tj be a sequence of m–rectifiable G chains, converging in flat norm to some T ∈ Rm(X,G). Fix ε > 0. According to the approximation theorem, there exist a P ∈ Pm(X,G) and F : X → X such that P = F#T +E with M(E) ≤ ε. Note that since F is Lipschitz, one has limj→∞ F(F#Tj −F#T ) = 0 and hence also limj→∞ F(E +F#Tj −P ) = 0 by the definitions of E and P . Since P is polyhedral, we can use Lemma 5.2.1:

(E + F#T ) = (P ) ≤ lim inf (E + F#Tj) ≤ M M j→∞ M m m ε + lim inf(Lip(F )) (Tj) ≤ ε + (1 + ε) lim inf (Tj), (5.11) j→∞ M j→∞ M where the last inequality follows from the fact that F is (1 + ε)–Lipschitz. On the other hand, since Lip(F −1) ≤ 1 + ε, we infer that

−m M(E + F#T ) ≥ M(F#T ) − M(E) ≥ (1 + ε) M(T ) − ε. (5.12) The lines (5.11) and (5.12) now give

−m m (1 + ε) (T ) ≤ (1 + ε) lim inf (Tj) + 2ε, M j→∞ M and the theorem will follow after we let ε tend to zero.

Corollary 5.2.1. Let (X, || ... ||) be an n–dimensional Banach space. If T ∈ Rn−1(X,G) and if πW : X → W is a W –good projection then the following inequality holds true:

M(πW #(T )) ≤ M(T ). Proof. Note that according to the strong approximation Theorem 5.2.2 there exists a sequence Pj ∈ Pn−1(X,G) such that F(T − Pj) → 0 and M(T ) = limj M(Pj). Next, since Lip(πW ) < ∞, we infer that F(πW (T − Pj)) → 0 and hence referring to Theorem 5.2.1 we write (πW #T ) ≤ lim inf (πW #Pj) ≤ lim inf (T ), M j M j M where the second inequality follows from Theorem 4.3.5.

5.3 Solution to the Plateau problem whose support stays inside of the convex hull of the boundary.

As a consequence of the lower semicontinuity theorem, we obtain the following existence result. Theorem 5.3.1. Let (X, || ... ||) be an n–dimensional Banach space and let G be a locally compact complete normed Abelian group. Then the following Plateau problem admits a solution: ( minimize M(T ) T ∈ Rn−1(X,G) such that ∂T = B, The “classical” proof of the existence of mass minimizing 66 (n − 1)–rectiﬁable G chains.

where B ∈ Rn−2(X,G) has a compact support and satisﬁes ∂B = 0. Moreover, among the solutions, there exists at least one, say T0 satisfying spt(T0) ⊆ conv(spt(B)).

Proof. We apply the direct method of calculus of variation. Take a minimizing sequence Tj ∈ Rn−1(X,G). By this we mean that ∂Tj = B and M(Tj) ≤ m + 1/j, where m is the inﬁmum of the problem. In what follows we are going to construct another minimizing 0 0 sequence Tj with an extra property: all spt(Tj) lie in one compact set. To this end, consider any bounded convex polyhedron Q such that B ⊂ Q. Denote by Q1,...,QN the (n − 1) dimensional facets of Q, by W1,...,WN the corresponding hyperplanes, by K the set N [ K = Q ∪ Wi i=1

+ + T + and finally by W1 ,...,WN the halfspaces in X satisfying Q = i Wi . Define mappings + ρi : X → X as ρi(x) = x if x ∈ Wi and ρi(x) = πWi (x), where πWi = GP(Wi), otherwise. Write ρ = ρ1 ◦ ... ◦ ρN . We claim that M(ρ#(Tj)) ≤ M(Tj). First, it is obviously sufficient to show that M(ρi#(Tj)) ≤ M(Tj) for each 1 ≤ i ≤ N. On the other hand,

+ + M(ρi#(Tj)) = M((Tj) (X\Wj )) + M(πWi#(Tj) (Wj )) ≤ M(Tj), where the last inequality here follows from Corollary 5.2.1. Next, it follows from the construction that spt(ρ#Tj) ⊂ K and that ∂(ρ#Tj) = B. Note we have obtained another minimizing sequence Tej = ρ#Tj that has support in K. Hence it is left to “get rid” of the S union of hyperplanes i Wi. Fix an integer i between 1 and N. Consider mappings fi : K → X defined as follows. First, fi(x) = x if x ∈ K\Wi and fi(x) = Pi(x) otherwise, where Pi : Wi → Qi is the Euclidean nearest point projection onto the convex set Qi. Let us prove that the functions fi are Lipschitz. Note that it is sufficient to estimate |fi(x) − fi(y)| when x ∈ Wi and y ∈ K\Wi. Note that by definition fi(y) = y and denote by z = fi(x). Suppose first that there holds ha, bi ≤ 0 where a = x − z, b = y − z. Hence

(|x − z| + |z − y|)2 (|f (x) − f (y)|)2 |x − y|2 = |ha − b, a − bi|2 ≥ |a|2 + |b|2 − 2ha, bi ≥ ≥ i i . 2 2

Hence we can assume that ha, bi ≥ 0. In this case there exists a constant C = Ci ∈ (0, 1) depending on Wi and Q such that

ha, bi 1 − C ≥ ≥ 0. |a||b|

Indeed, this follows from the fact that the angles between the facets Qj, j 6= i and Wi are 5.3 Solution to the Plateau problem whose support stays inside of the convex hull of the boundary. 67 bounded from below. Now we estimate:

|x − y|2 ≥ |a|2 + |b|2 − 2(1 − C)|a||b| = C2 C2 C 2 |a|2 + |b|2 + 2|a||b| + 1 − |a|2 + |b|2 − 2 1 − |a||b| ≥ 4 4 2 C2 C2 (|x − z| + |z − y|)2 ≥ |f (x) − f (y)|2, 4 4 i i where the second inequality here follows from the fact that D := (1−C/2)4−(1−C2/4)2 ≤ 0 when C ∈ (0, 1). Next, extend according to the Kirszbraun theorem 3.2.8 the functions fi to the whole space X preserving the Lipschitz condition. The Lipschitz constants of fi may grow, but this is not important. We call the extended functions fi as well. Consider further mappings gi is deﬁned as the Euclidean nearest point projections gi : X → Qi onto the facet Qi. Consider now the chain fi#Tej. Note that spt(fi#Tej) ⊆ S Q ∪ k6=i Wk. In order to estimate its mass, write

N X Tej = R0 + Rk, k=1 where Rk = Tej (Wk \ Q) and R0 = Tej Q. Note that spt(Rk) ⊂ Wk and that fi = gi on Wi and hence fi#Ri = gi#Ri. On the other hand, there exists a constant C = C(Wi) such that M(A) = CM2(A) (where M2 stands for the Euclidean mass) for all chains A such that spt(A) ⊂ Wi. Hence, since Lip(gi) ≤ 1 we infer that

M(fi#Ri) = CM2(gi#(Ri)) ≤ CM2(Ri) = M(Ri).

We ﬁnally estimate the mass of the chain fi#Tej:

N N N X X X M(fi#Tej) ≤ M(fi#Rk) = M(Rk) + M(fi#Ri) ≤ M(Rk) = M(Tej). k=0 k6=i k=0 Performing the same procedure for each i between 1 and N consequently, we obtain a 0 0 minimizing sequence Tj,Q such that spt(Tj,Q) ⊆ Q. Write C = conv(spt(B)). According T + to the Hahn–Banach theorem we know that C = x∈∂C Hx , where Hx is a supporting + hyperplane to C and Hx is the corresponding halfspace. Let us prove that ∞ \ + C = Hi i=1

+ ◦ ∗ for a countable subset of halfspaces Hi . Indeed, the polar set C = X ∩ {α : supC α ≤ 1} ◦ of the convex set C is separable, hence there exists a dense sequence αn in C . Note that \ C = α−1(−∞, 1]. α∈C◦ The “classical” proof of the existence of mass minimizing 68 (n − 1)–rectiﬁable G chains.

T −1 We infer that C = n αn (−∞, 1]. Obviously, C is contained in the intersection. We prove T −1 the reciprocal inclusion by contradiction. Take c∈ / C, but c ∈ n αn (−∞, 1]. Hence there ◦ exists α ∈ C and x ∈ C such that α(x) ≥ 1 + ε for some ε > 0. Approximating α by αn with n large enough yields a contradiction. Next write

N \ + CN = Hi i=1 with some N such that CN is bounded. Apply the procedure described in the previous three paragraphs to the polyhedron C . Note that T 0 ∈ F (X,G) ∩ {T : (T ) ≤ N j,CN n−1 N C, supp(T ) ⊆ CN }, which is F–compact by Theorem 3.2.7. Hence there exists TN ∈ n−1 0 R (X,G) such that Tj,N → TN . Since the mass is lower semicontinuous, we infer that 0 1 M(TN ) ≤ lim inf M(Tj,N ) ≤ lim inf m + = m, j j i

and hence M(TN ) = m. After a subsequence TNk → T0 and obviously ∂T0 = B and T M(T0) = m. On the other hand, spt(T0) ⊆ k CNk = C. Chapter 6

Gross measures and the second proof of the existence in codimension one.

In this chapter we are going to present our second proof of the existence of mass minimising (n − 1)–rectiﬁable G chains in n–dimensional Banach spaces.

Remark 6.0.1. The theorems proved in this chapter were ﬁrst formulated and proved here by I.Vasilyev.

6.1 A little bit more about good projections.

The current section is the one where we establish a couple of useful properties of good projections.

Remark 6.1.1. Let X be a Banach space. We denote here by Hom(X,X) the space of bounded linear operators on X furnished with the norm topology (for f ∈ Hom(X,X) we deﬁne |||f||| := sup{||f(x)|| : x ∈ X, ||x|| = 1}).

Proposition 6.1.1. The set Pr of all pairs (W, πW ), where W ∈ G(n − 1,X) and π is a W –good projection, is a closed subset of the space G(n − 1,X) × Hom(X,X), furnished with the product topology.

∞ Proof. Let {Wi}i=1 be a sequence of hyperplanes in X and for each i ≥ 1 let πWi be a

Wi–good projection. Suppose that (Wi, πWi ) → (W, πW ) in G(n − 1,X) × Hom(X,X). Then the goal is to prove that πW is a W –good projection. Let V ∈ G(n − 1,X) and let n−1 n−1 A ⊂ V . Note that we are done once we prove that H||...||(πW (A)) ≤ H||...||(A). Since each n−1 n−1 πWi is Wi–good, we know that H||...||(πWi (A)) ≤ H||...||(A). Next, thanks to Theorem 4.3.1, we know that

n−1 n−1 n−1 H (πWi (A)) n−1 H (A) H||...||(πWi (A)) = αn−1 n−1 and H||...||(A) = αn−1 n−1 . H (B ∩ Wi) H (B ∩ V ) Gross measures and the second proof of the existence in 70 codimension one.

This allows us to conclude that

n−1 n−1 H (πWi (A)) H (A) n−1 ≤ n−1 . H (B ∩ Wi) H (B ∩ V ) Hence it is suﬃcient to establsih the following two convergences:

n−1 n−1 n−1 n−1 H (πWi (A)) → H (πW (A)) and H (B ∩ Wi) → H (B ∩ W ).

The second convergence can be derived from Theorem 4.2.1. Indeed, this theorem tells that the Busemann–Hausdorff density function is convex, hence it is also continuous (of course one should also use an obvious fact that Hn−1(B ∩Wf) 6= 0 for all Wf ∈ G(n−1,X)). Let us prove the first convergence. Let π : V → W be a mapping defined as the gWi restriction of the projection πWi to the hyperplane V . Note that

Hn−1(π (A)) = J π ·Hn−1(A), Wi gWi where J π is the Jacobian of the mapping π . Fix w , . . . , w , orthonormal bases gWi gWi i,1 i,n−1 of Wi. Note that we can suppose that the bases wi,1, . . . , wi,n−1 converge towards some basis v1, . . . , vn−1 of the hyperplane V . Indeed, we can consequently, for j from 1 to n − 1, extract subsequences wj,i converging to some vj. Since the scalar product is a continuous operation the limiting system v1, . . . , vn−1 will be an orthonormal basis of V . This might force us to consider a subsequence of the sequence Wi, but this changes nothing. Then for all indexes 1 ≤ i and 1 ≤ j ≤ n − 1 one has

n−1 X πWi (vj) = ai,j,kwi,k k=1 for some real coeﬃcients ai,j,k. On the other hand, we know that

hvj, ωii πWi (vj) = vj − ui, hvj, uii where ωi is the Euclidean unit normal to the hyperplane Wi and ui is the Euclidean unit vector, spanning the (one–dimensional) kernel of the projection πWi . Hence for all indexes 1 ≤ k ≤ n − 1 one has

hvj, ωii ai,j,k = hπWi (vj), wi,ki = hvj, wi,ki − hui, wi,ki. hvj, uii

Since J π = det{a }n−1 , the desired convergence follows. gWi i,j,k k,j=1 In what follows we are going to use the following two deﬁnitions.

Deﬁnition 6.1.1. (Polish space) A Polish space is a separable topological space that can be metrized using a complete metric. 6.2 The Gross measures and another proof of the existence in codimension one. 71

Deﬁnition 6.1.2. (Suslin set) A subset of a Polish space X is an analytic set if it is a continuous image of some Polish space.

The following result can now be derived from Theorem 6.1.1.

Corollary 6.1.1. Let (X, || ... ||) be an n–dimensional Banach space. Then there exists a universally measurable mapping GP : G(n − 1,X) → Hom(X,X) such that for every W ∈ G(n − 1,X), GP(W ) is a W –good projection. We shall often write πW instead of GP(W ). We shall call such mapping GP a good choice of good projections mapping.

Proof. Since every closed set is Suslin (analytic), the corollary follows from [Coh13], The- orem 8.5.3.

6.2 The Gross measures and another proof of the existence in codimension one.

Following Federer ([Fed69], section 2.10.3) we deﬁne the Gross measures.

Deﬁnition 6.2.1. (Gross measure) Let P : G(n − 1,X) → Hom(X,X) be a map such that for every W ∈ G(n − 1,X) the mapping P(W ) is a linear projection with range W . For a Borelian set A ⊆ X denote

n−1 n−1 ζ||...||,P(A) := sup H||...||(PW (A)), W ∈G(n−1,X)

n−1 The Gross measure G||...||,P is deﬁned by the following formula

( ∞ ∞ ) n−1 X n−1 [ G||...||,P(S) := sup inf ζ||...||,P(Aj): Aj ⊂ X are Borelian, Aj ⊃ S, diam(Aj) ≤ δ . δ>0 j=1 j=1

n−1 Remark 6.2.1. Notice that the measure G||...||,P is deﬁned by the Caratheodory construction using a gauge deﬁned on Borelian subsets of X. Hence it is automatically a Borel regular measure, see [Fed69] 2.10.4.

The following theorem shows that for rectifiable subsets of finite dimensional Banach spaces the Hausdorff and the Gross measures coincide.

Theorem 6.2.1. Let (X, || ... ||) be an n–dimensional Banach space and let GP : G(n − 1,X) → Hom(X,X) be a good choice of good projections mapping. If A is a Borelian (n − 1)–rectiﬁable subset of X, then

n−1 n−1 H||...||(A) = G||...||,GP(A). Gross measures and the second proof of the existence in 72 codimension one.

n−1 n−1 Proof. Let us ﬁrst prove that H||...||(A) ≥ G||...||,GP(A). Fix δ > 0. Let {Bj}j∈J be a S sequence of Borelian sets satisfying A = j Bj and diam||...||(Bj) ≤ δ and such that the sets Bj are pairwise disjoint. Indeed, one can take√ Bj = Cj ∩ A, where Cj is the system of the dyadic semi–cubes with edges of length δ/ n. Next, since each πW = GP(W ) is a W –good projection, thanks to Theorem 4.3.4 for each j ∈ J we have

n−1 n−1 ζ||...||,GP(Bj) ≤ H||...||(Bj).

n−1 Let us take the sum in the previous line and use the fact that G||...||,GP is a Borelian measure:

n−1 X n−1 X n−1 n−1 G||...||,GP(A) ≤ ζ||...||,GP(Bj) ≤ H||...||(Bj) = H||...||(A), j∈J j∈J and the first inequality follows. We proceed to the reciprocal inequality. Note that since A is (n − 1)–rectifiable, there 1 n−1 exists a countable union of C –manifolds {Mi}i∈I and a set M0 such that H (M0) = 0 S S satisfying A ⊆ i Mi ∪ M0. Writing A as a disjoint union A = i Ai ∪ M0 where Ai ⊆ Mi are Borelian, we infer that it is sufficient to prove the inequality for Borelian subsets of C1–manifolds. Let A ⊆ M be a Borelian subset of manifold M. Fix ε > 0. First, we claim that for every x ∈ A there exists δ = δ(x) such that for all r < δ there holds

n−1 n−1 H||...||(A ∩ B(x, r)) ≤ (1 + ε)H||...|| (πW (A ∩ B(x, r))) , where by W we denote the tangent plane Tan(M, x) to the manifold M at the point x and πW = GP(W ). Indeed note that we are done once we show that the function πW (A ∩ B(x, r)) is (1 + ε)–Lipschitz with respect to the norm || ... ||. To this end, write πW (y) = y + ωg(y), where ω is the Euclidean unit vector, spanning the (one–dimensional) kernel of the projection πW and g is a real valued function. Hence

||πW (y) − πW (z)|| ≤ ||y − z|| + ||ω|| · |g(y) − g(z)| and it remains to show that the function g : A ∩ B(x, r) → R is ε–Lipschitz for r small enough. To prove this, consider the orthogonal projection PW : X → W given by the formula PW (y) = y + ef(y), where e is the Euclidean unit normal vector to the hyperplane W and f is a real valued function. Since M is a manifold, it follows that there exists δ = δ(x) such that for all r ≤ δ the function f (M ∩ B(x, r)) is ε–Lipschitz. Note that for all z, y ∈ X one has g(z)f(y) = g(y)f(z). Hence for all z, y ∈ M ∩ B(x, r) and all r ≤ δ one has

g(z) g(y) |g(y) − g(z)| ≤ |g(y)| · 1 − ≤ · |f(y) − f(z)| ≤ Cε||y − z||, g(y) f(y) and our claim follows. 6.2 The Gross measures and another proof of the existence in codimension one. 73

Second, we infer that for every Borelian subset E ⊆ X there holds

n−1 n−1 ζ||...||(E) ≤ G||...||(E). (6.1) S Indeed, suppose E ⊆ j∈J Bj, where Bj are Borelian. For each V ∈ G(n − 1,X) one obviously has n−1 X n−1 H||...||(πV (E)) ≤ H||...||(πV (Bj)), j∈J where πV = GP(V ). Hence

n−1 n−1 X n−1 ζ||...||(E) = sup H||...||(πV (E)) ≤ sup H||...||(πV (Bj)) ≤ V ∈G(n−1,X) V ∈G(n−1,X) j∈J X n−1 X n−1 sup H||...||(πV (Bj)) = ζ||...||(Bj). j∈J V ∈G(n−1,X) j∈J

Since {Bj}j∈J is arbitrary, the inequality follows. Finally, consider the following collection of balls B = {B(x, r): x ∈ A, 0 < r < δ(x)} which is a covering of the set A. Referring to the Besicovitch covering theorem 3.2.9 we infer that there exists a countable disjoint collection of balls {B(xi, ri)}i∈I ⊂ B such that n−1 S H||...||(A\( i B(xi, ri))) = 0. Hence we can estimate

n−1 X n−1 X n−1 H||...||(A) = H||...|| (A ∩ B(xi, ri)) ≤ (1 + ε) ζ||...|| (A ∩ B(xi, ri)) ≤ i∈I i∈I X n−1 n−1 (1 + ε) G||...|| (A ∩ B(xi, ri)) ≤ (1 + ε)G||...|| (A) , i∈I and the theorem follows. Remark 6.2.2. We would like to emphasise that in the previous theorem the first inequal- n−1 n−1 ity (namely H||...||(A) ≥ G||...||,GP(A)) does not require the rectifiability of A. Next we give one more definition in connection with the Gross measures. Definition 6.2.2. (Gross mass) Let (X, || ... ||) be an n–dimensional Banach space and let G be a complete normed Abelian group. Suppose that T ∈ Rm(X,G) with T = m θT (x)H set(T ). We define the Gross mass MG by the following formula Z n−1 MG(T ) := θT (x)dG||...||,GP(x). set(T ) We shall use Theorem 6.2.1 in our second proof of the lower semicontinuity result in codimension one.

Theorem 6.2.2. Let (X, || ... ||) be an n–dimensional Banach space. Then the mass M is lower semicontinuous with respect to ﬂat convergence on Rn−1(X,G). Gross measures and the second proof of the existence in 74 codimension one.

n−1 Proof. Deﬁne an auxiliary mass denoted as ζ||...||,GP and deﬁned for T ∈ Rn−1(X,G) as

n−1 ζ||...||,GP(T ) := sup M(πW #(T )), W ∈G(n−1,X) where πW = GP(W ). Note that, thanks to Theorem 6.2.1, it is sufficient to show that the Gross mass MG is lower semicontinuous with respect to flat convergence. The rest of the proof is devoted to this statement. Let us trace briefly the plan of our proof. First, we prove a lemma where n−1 we show that the mass ζ||...||,GP is itself F–lower semicontinuous. Second, we prove that n−1 the mass ζ||...||,GP is majorized by the mass MG (by the way, this step requires a variant of Crofton’s formula). We finally derive the desired lower semicontinuity from the first two steps using a covering argument which depends on a formula for the Gross density.

Lemma 6.2.1. If {Ti} is a sequence of (n−1)–rectiﬁable G chains such that F(Ti −T ) → 0 for some T ∈ Rn−1(X,G), then

n−1 n−1 ζ (T ) ≤ lim inf ζ (Ti). ||...||,GP i ||...||,GP

Proof. Given W ∈ G(n − 1,X), since πW is Lipschitz, one has F(πW #T − πW #Ti) → 0. Note that on the space Rn−1(W, G) ﬂat norm F is equivalent to the mass M. Now we estimate:

lim | (πW #T ) − (πW #Ti)| ≤ lim (πW #T − πW #Ti) ≤ i M M i M

C(n) lim F(πW #T − πW #Ti) ≤ C(n) lim F(T − Ti). i i

Hence for each W the application T → M(πW #(T )) is continuous. Since supremum of continuous functions is lower semicontinuous, the proof is complete.

In what follows we will need a variant of Crofton’s inequality, recall 2.2. The desired version resembles the inequality from Proposition 5.5.1 in [BP15]. In order to prove this result, we ﬁrst give a number of deﬁnitions from [BP15].

Deﬁnition 6.2.3. (Multiplicity function) Given A ⊂ X, its multiplicity function NA is deﬁned as follows

NA : G(n − 1,X) × X → R+ ∪ {∞} −1 (W, y) 7→ card(A ∩ πW (y)), where πW = GP(W ). Deﬁnition 6.2.4. (Universally measurable set) Given a Polish space Z (see 6.1.1) with Borelian sigma–algebra Ω, a subset A ⊂ Z is called a universally measurable set if it lies in the completion of the σ–algebra Ω with respect to every ﬁnite measure µ on (Z, Ω). 6.2 The Gross measures and another proof of the existence in codimension one. 75

Deﬁnition 6.2.5. (Universally measurable function) We call a mapping f : Y → Z where Y and Z are two measurable spaces a universally measurable function if f −1(A) ⊂ Y is a universally measurable set, once the set A ⊂ Z is Borelian. Remark 6.2.3. It is worth mentioning that universally measurable sets form a σ–algebra. With each set A ⊂ X we associate the set χ(A) deﬁned as

χ(A) = {(W, y) ∈ G(n − 1,X) × X : y ∈ πW (A)}.

We also ﬁx the following notation: denote by Qj a system of dyadic semi–cubes with edges of the length 2−j. Let us prove the following lemma which gives a formula for the function NA.

Lemma 6.2.2. Let A ⊂ X and let Bj = {Q ∈ Qj : A ∩ Q 6= ∅}. It follows that X NA = lim 1χ(B). j B∈Bj Proof. The proof is identical to that of the Proposition 5.4.1 of the paper [BP15]. Nev- ertheless we present here a proof for the reader’s convenience. Fix (W, y) ∈ G(X, m) × X. For each integer j define Gj = Bj ∩ {B : 1χ(B)(W, y) = 1}. Next, one easily defines an injective map Gj → Gj+1. Thus the sequence NA,j(W, y) is nondecreasing and therefore has a limit in R+ ∪ {∞}. Our next claim is that NA(W, y) ≥ NA,j(W, y) for every integer j. Let k ∈ Z+ be such that NA,j(W, y) ≥ k. This means that there are distinct B1,...,Bk ∈ Bj such that (W, y) ∈ χ(Bk), k = 1, . . . k. In particular there are −1 xk ∈ Bk so that y = πW (xk), therefore also xk ∈ A ∩ πW (y). Since the sets Bk are distinct so are the xk and we conclude that NA(W, y) ≥ k. Since k is arbitrary the proof of the claim is complete. In order to finish the proof of the proposition it remains to show that

NA(W, y) ≤ lim NA,j(W, y). j

−1 0 0 Let F ⊆ A ∩ πW (y) be a ﬁnite set. Put δ := min{|x − x | : x, x ∈ F are distinct}. Making the side lengths of the cubes small enough, we infer that there is an integer j0 such that diam(B) < δ whenever B ∈ Bj and j ≤ j0. Fix such j. Then each x ∈ F belongs to 0 some B ∈ Bj and two distinct x, x ∈ F cannot belong to the same B ∈ Bj . Therefore −1 NA,j(W, y) ≤ card(F ). If NA(W, y) < ∞ then we simply let F = A∩πW (y) and the lemma is established. Otherwise the argument shows that limj NA,j(W, y) = ∞ and the assertion of the lemma follows in this case as well. We further infer that given a hyperplane W ∈ G(n−1,X) and a universally measurable set A the function y → NA(W, y) Gross measures and the second proof of the existence in 76 codimension one. is universally measurable. Indeed, this follows from the previous lemma and from the fact that the set πW (B) is universally measurable for each B ∈ Bj, since πW is continuous. As a consequence of this fact we observe that the function

gA : G(n − 1,X) → R+ ∪ {∞} Z n−1 W 7→ NA(W, y)d H||...|| W (y) X is well deﬁned. Now we are ready to state the needed variant of Crofton’s inequality: Theorem 6.2.3. Let A ⊂ X be a Borelian set. Then the following holds

n−1 sup gA(W ) ≤ G||...||,GP(A). W ∈G(n−1,X)

Proof. Let Bj be as above. Thus by Lemma 6.2.2 we infer that X NA(W, y) = lim 1χ(B)(W, y) j B∈Bj for every (W, y) ∈ G(n − 1,X) × X. Thanks to the Monotone Converge Theorem, for each W we have Z n−1 gA(W ) = NA(W, y)d H||...|| W (y) = X Z X n−1 X n−1 lim 1χ(B)(W, y)d H W (y) = lim H (πW (B)). j ||...|| j ||...|| B∈Bj X B∈Bj Next, referring to basic properties of the supremum we get

X n−1 sup gA(W ) ≤ lim inf sup H||...||(πW (B)) = W ∈G(n−1,X) j W ∈G(n−1,X) B∈Bj X lim inf ζn−1(B) ≤ Gn−1(A), j ||...|| ||...|| B∈Bj where the last inequality here follows from the inequality (6.1). As a consequence of this theorem we obtain the following proposition.

Proposition 6.2.1. Let θ : X → R+ be a Borelian function. Then for every W ∈ G(n − 1,X) holds Z Z X n−1 n−1 θ(y)d(H W )(x) ≤ θdG||...||,GP. −1 X y∈πW (x) X 6.2 The Gross measures and another proof of the existence in codimension one. 77

Proof. Note that the case θ = 1A corresponds to the previous theorem. Next, if θ = P 1 j∈J aj Aj , then we use the additivity of the integral and the fact that sup(f + g) ≤ sup(f) + sup(g). The general case (i.e. when θ is Borel) is treated via regarding a nondecreasing sequence of step functions converging pointwise to the function θ and using the Monotone Convergence Theorem.

n−1 We continue the proof of Theorem 6.2.2 with a lemma, which connects MG and ζ||...||,GP. n−1 Lemma 6.2.3. For any T ∈ Rn−1(X,G) there holds ζ||...||,GP(T ) ≤ MG(T ). Proof. Note that for each W ∈ G(n − 1,X) one has X θπW #(T )(x) ≤ θT (y). −1 y∈πW (x)∩set(T )

Hence Z X n−1 M(πW #(T )) ≤ θT (y)dH||...||(x). −1 W y∈πW (x)∩set(T ) Now we take the supremum over all W ∈ G(n − 1,X) and further use Proposition 6.2.1 in order to conclude that Z n−1 n−1 ζ||...||,GP(T ) ≤ θT (x)dG||...||,GP(x) ≤ MG(T ), X and the lemma follows. We shall need one more lemma that gives a formula for the Gross density. n−1 Lemma 6.2.4. Suppose that T ∈ Rn−1(X,G). Then for H –almost all x ∈ set(T ) one has n−1 ζ||...||,GP(T B(x, r)) lim n−1 = θT (x). r→0 αn−1r Proof. First note that according to Lemma 6.2.3 one has:

ζn−1 (T B(x, r)) ||...||,GP MG(T B(x, r)) lim n−1 ≤ lim n−1 = r→0 αn−1r r→0 αn−1r Z 1 n−1 lim n−1 θT (y)dH||...||(y) = θT (x). r→0 αn−1r B(x,r)∩set(T )

Hence it is left to prove the reciprocal inequality. In what follows we shall require a notion of approximately continuous functions. We refer the reader to the book [EG15] for the deﬁnition and properties of these functions. Suppose ﬁrst that set(T ) ⊆ M where M is a smooth manifold. Take x ∈ set(T ) such that θT is approximately continuous at x 1 and denote W0 = Tan(M, x). Indeed, since θT ∈ L (X,G) it is approximately continuous Gross measures and the second proof of the existence in 78 codimension one.

n−1 H M–almost everywhere. Fix ε > 0. There exists ρ1 = ρ1(x) such that for all r ≤ ρ1(x) there holds n−1 H (πW (M ∩ B(x, r))) ||...|| − 1 ≤ ε. n−1 αn−1r Indeed, this follows from the proof of Theorem 6.2.1 and also from the fact that x satisﬁes n−1 n−1 Θ||...||(H||...|| M, x) = 1. Next we choose ρ2 such that M ∩B(x, ρ2) = F (πW (M ∩B(x, ρ2))) −1 where πW = GP(W ) and F = πW (M ∩ B(x, ρ2)). Since θ is approximately continuous at x we can choose a subset Er ⊂ M ∩ B(x, r) such that θT (y) ≤ θT (x) − ε for all y ∈ Er n−1 n−1 and H||...||(Er) ≤ εαn−1r . Now for r ≤ min{ρ1, ρ2} we estimate

n−1 ζ||...||,GP(T B(x, r)) n−1 ≥ αn−1r Z 1 −1 n−1 n−1 θT (πW (M ∩ B(x, r))(y)dH||...||(y) ≥ αn−1r πW (M∩B(x,r))\Er n−1 n−1 θT (x)H||...||(πW (M ∩ B(x, r))\Er) εH||...||(πW (M ∩ B(x, r))) n−1 − n−1 ≥ αn−1r αn−1r n−1 n−1 θT (x) H||...||(πW (M ∩ B(x, r))) − εαn−1r n−1 − 2ε ≥ αn−1r

θT (x)(1 − ε) − 3ε.

Let us now treat the general case. Write

∞ X T = Ti, i=1 where Ti = T Ai, with Ai ⊆ Mi– Borelian subsets of smooth manifolds. Note that for Hn−1 set(T ) almost all x there exists k ∈ N such that there holds

n−1 n−1 n−1 n−1 Θ||...||(H||...|| Ak, x) = 1 and Θ||...||(H||...|| Aj, x) = 0, if j 6= k, j ∈ N. Hence for such x there holds

n−1 n−1 ζ||...||,GP(T B(x, r)) ζ||...||,GP(Tk B(x, r)) n−1 − n−1 ≥ αn−1r αn−1r ζn−1 ((T − T ) B(x, r)) ||...||,GP k M((T − Tk) B(x, r)) − n−1 ≥ − n−1 . αn−1r αn−1r Taking the limit with respect to r → 0 in the last line yields the desired inequality and ﬁnishes the proof of the lemma. 6.2 The Gross measures and another proof of the existence in codimension one. 79

We are now ready to ﬁnish the proof of the announced lower semicontinuity result. Fix n−1 ε > 0. For H –almost every x ∈ set(T ) there exists ρx > 0 such that n−1 MG(T B(x, r)) ≤ (1 + ε)ζ||...||,GP(T B(x, r)) (6.2) for all 0 < r ≤ ρx. This follows from the previous lemma and from B.Kirchheim’s theorem 3.2.3 that −1 −(n−1) n−1 lim αn−1r H (set(T ) ∩ B(x, r)) = 1 r→0 ||...|| n−1 for H –almost all x ∈ set(T ). Indeed, taking once again a point x ∈ set(T ) such that θT is approximately continuous at x we infer that

n−1 MG(T B(x, r)) ≤ (ε + θT (x))H||...||(set(T ) ∩ B(x, r)) ≤ 2 n−1 n−1 (1 + εe) αn−1r θT (x) ≤ (1 + εb)ζ||...||,GP(T B(x, r)), where εb → 0 when ε → 0. Moreover, for each such x almost all 0 < r < ρx have the additional property that limr→0 F(T B(x, r) − Ti B(x, r)) = 0 according to [PH12], 5.2.3. The collection B of balls having all the stated properties is a Vitali covering of a conegligible subset of set(T ) (i.e. a subset of set(T ) whose complement in set(T ) has Hn−1 measure zero). According to the Besicovitch covering theorem 3.2.9, there exists a disjoint n−1 countable subcollection {B(xj, rj): j ∈ J} of B that covers H –almost all of set(T ). Hence we can estimate using the inequality (6.2):

X X n−1 MG(T ) = MG(T B(xj, rj)) ≤ (1 + ε) ζ||...||,GP(T B(xj, rj)) = j∈J j∈J X n−1 = (1 + ε) lim inf ζ (Ti B(xj, rj)) ≤ ... i→∞ ||...||,GP j∈J where the last equality follows from Lemma 6.2.1. We ﬁnish the estimate, now using Lemma 6.2.3: X X ... ≤ (1 + ε) lim inf G(Ti B(xj, rj)) ≤ (1 + ε) lim inf G(Ti B(xj, rj)) ≤ i→∞ M i→∞ M j∈J j∈J

≤ lim inf G(Ti), i→∞ M where the last equality follows from the fact that the balls B(xi, ri) are pairwise disjoint. Hence the theorem follows. As a consequence of our lower semicontinuity theorem, we obtain the following existence result. Theorem 6.2.4. Let (X, || ... ||) be an n–dimensional Banach space and let G be a locally compact complete normed Abelian group. Then the following Plateau problem admits a solution ( minimize M(T ) T ∈ Rn−1(X,G) such that ∂T = B, Gross measures and the second proof of the existence in 80 codimension one.

where B ∈ Rn−2(X,G) has a compact support and satisﬁes ∂B = 0. Moreover, among the solutions, there exists at least one, say T0 satisfying spt(T0) ⊆ conv(spt(B)). Proof. It is suﬃcient to recall Theorem 5.3.1. Chapter 7

The existence in the two dimensional case.

We proceed to the sixth chapter, where we will establish the existence of mass minimising 2 rectiﬁable G chains in ﬁnite dimensional Banach spaces.

7.1 Some preliminary deﬁnitions and remarks.

We start with a number of motivating definitions and remarks. Definition 7.1.1. (Totally convex densities) We call a k–density function φ on an n– dimensional Banach space X, n > k totally convex if for every k–dimensional linear subspace there exists a linear projection onto that subspace which doest not increase the volume Volφ corresponding to the density φ, see 4.1.1. There exists an equivalent way of defining totally convex densities. We mean the following result discussed in [TA04]. Theorem 7.1.1. A k–density φ on an n–dimensional vector space X, n > k, is totally convex if and only if it is extendibly convex and moreover the following holds. If Φ : Λk(X) → R is a convex extension of φ, then through every point of the unit sphere S = {x ∈ Λk(X) : Φ(x) = 1} there passes a supporting hyperplane of the form ξ = 1, ∗ where ξ a simple k–vector in Λk(X ). Remark 7.1.1. Note that, thanks to Theorem 4.3.2, in case when m = n − 1 each Busemann–Hausdorff density is totally convex. The following remark shows that there exist densities which are convex but not totally convex. However, the author does not know, whether there exists a Busemann–Hausdorff density, that is not extendibly convex. Remark 7.1.2. For m 6= {1, n − 1} it is possible to construct an example of a m–volume density which is extendibly convex, but not totally convex (see [BS60]). The density, constructed in [BS60] is called the quadratic density. 82 The existence in the two dimensional case.

Hence it seems natural to me that one should use an approach that is diﬀerent from the linear projections machinery, developed in the third and forth chapters, while trying to attack the Plateau problem in the codimensions diﬀerent from 1 and (n − 1). In the case of codimension (n − 2) it is possible to provide another successful method based on one theorem, proved in the paper [BI12] by Burago and Ivanov.

7.2 Burago–Ivanov’s theorem and some of its consequences.

We would like to emphasize that throughout this chapter we denote by φgBH a function acting from the Grassmann cone GC(2,X) to R+ and deﬁned for a simple 2–vector σ = v1 ∧ v2 as φgBH (σ) = |σ|2 · φBH (span{v1, v2}). The main result of this section is the following theorem which is nothing but the “triangle inequality” for the 2–cycles.

Theorem 7.2.1. (I.Vasilyev) Let (X, || ... ||) be an n–dimensional Banach space with the unit ball B = {x ∈ X : ||x|| ≤ 1}. Let P ∈ P2(X,G) be a 2–cycle (i.e. ∂P = 0) such that

N X P = gj[σj], j=1 where σj are non–overlapping 2–simplexes. Then

N 2 X 2 |g1|H||...||(σ1) ≤ |gj|H||...||(σj). j=2

Remark 7.2.1. This result was ﬁrst formulated and proved in the present thesis by I.Vasilyev.

Our proof of Theorem 7.2.1 highly depends on the following result of Burago and Ivanov.

Theorem 7.2.2. (Burago–Ivanov) Let (X, || ... ||) be an n–dimensional Banach space. Suppose that the unit ball B = {x ∈ X : ||x|| ≤ 1} is a bounded convex polyhedron. For every W ∈ G(2,X) let M = MW be the polygon defined as the intersection B ∩ W , M = conv{a1, . . . , ad} and let Fi be a linear supporting function to the set B such that Fi equals 2 2 to one on the segment [ai, ai+1]. Define coefficients pi by pi = 2H (40aiai+1)/H (B ∩ W ) and a function α = αW :Λ2(X) → R+ as X α(σ) = π pkplhFk ∧ Fl, σi. (7.1) 1≤k

Then for all σ ∈ GC2(X) the function α satisﬁes

α(σ) ≤ φgBH (σ), (7.2) and if σ = v1 ∧ v2, with v1, v2 ∈ W linearly independent vectors, then

α(σ) = φgBH (σ). (7.3)

Proof. We shall first prove our Theorem 7.2.1. For each j from 1 to N, let Wj be the plane such that σj ⊂ Wj. Let us also for each j fix vj,1, vj,2, an orthonormal basis of Wj and further define ωj ∈ Λ2(X) as ωj = vj,1 ∧ vj,2. We claim that it is sufficient to prove the theorem in case when B is a bounded convex polyhedron. Indeed, we first note that for every ε > 0 and every j ∈ [1,...,N] there exists a 2–dimensional bounded convex polyhedron Pε,j satisfying Pε,j ⊆ B ∩ Wj and 2 H ((B ∩ Wj)\Pε,j) ≤ ε. Next we define by Pε the polyhedron, obtained as the convex combination of Pε,j, N ! [ Pε = conv Pε,j j=1 and by || ... ||ε the corresponding norm. Note that the norm || ... ||ε is well defined. Indeed, sym we can suppose that Pε is symmetric by substituting it with its symmetrization Pε if sym needed. Since B is symmetric we will have Pε ⊆ Pε ⊆ B and hence Pε ∩ Wj ⊆ B ∩ Wj 2 and H ((B ∩ Wj)\(Pε ∩ Wj)) ≤ ε. In principle, Pε can be of dimension n1 < n, but this changes nothing. Now we estimate

N N N 2 X 2 X 2 X H (σj) |gj|H (σj) = |gj|H (σj)φgBH (Wj) = |gj| ≥ ||...|| H2(B ∩ W ) j=2 j=2 j=2 j N 2 N 2 ! X H (σj) X H (σj) Cf1ε 2 |gj| ≥ |gj| 1 − ≥ |g1|H (σ1) − Cεe ≥ ..., H2(P ) + ε H2(P ) H2(P ) ||...||ε j=2 ε,j j=2 ε,j ε,j where the constants Ce and Cf1 do not depend on ε. On the other hand, H2(σ ) H2(σ ) |g |H2 (σ ) = |g | 1 ≥ |g | 1 . 1 ||...||ε 1 1 2 1 2 H (Pε,1) H (B ∩ W1) We ﬁnish the estimate:

2 H (σj) 2 ... ≥ |g1| 2 − Cεe = |g1|H||...||(σ1) − Cε,e H (B ∩ W1) and our claim will follow after letting ε tend to 0. Therefore we can assume that B is a bounded convex polyhedron. Let us use the theorem by Burago and Ivanov and keep the notations from there. Let α = αW1 be the 84 The existence in the two dimensional case.

function corresponding to the plane W1 (the one constructed in Theorem 7.2.2). Using the deﬁnition of the Hausdorﬀ–Busemann density, the inequality (7.2) and the formula (7.1) we write:

N N X 2 X 2 |gj|H||...||(σj) = |gj|H (σj)φgBH (ωj) ≥ j=2 j=2 N N X 2 X X 2 |gj|H (σj)αW1 (ωj) = |gj|H (σj)pkplhFk ∧ Fl, ωji = .... (7.4) j=2 j=2 1≤k

2 Following Burago and Ivanov, we deﬁne for all k < l from 1 to d mappings Fk,l : X → R by the formula Fk,l(x) = (Fk(x),Fl(x)). It follows now from the paper [BI12] (page 636 lines 1–3) that

2 2 H (σj)hFk ∧ Fl, ωji = H (Fk,l(σj)) (7.5) for all k < l from 1 to n and all j from 1 to N. This identity is an easy consequence of 2 the area formula. Indeed, the area H (Fk,l(σj)) of the image of the linear mapping Fk,l is 2 equal to the area the pre–image H (σj) times its Jacobian, which is equal to the absolute value of the determinant of the matrix

Fk(v1) Fl(v1) Fk(v2) Fl(v2).

Next, note that since ∂P = 0, polyhedral 2 chain Q = Fk,l#(P ) satisﬁes ∂Q = 0. Since Q has compact support, the constancy theorem tells us that Q = 0 and hence

N X M(g1[σ1]) ≤ M(gj[σj]). j=2 Using this and the line (7.5) we continue the estimate (7.4):

N N X X 2 X X 2 ... = |gj|pkplH (Fk,l(σj)) = pkpl |gj|H (Fk,l(σj)) ≥ j=2 1≤k

2 X 2 ... = |g1|H (σ1) pkplhFk ∧ Fl, ω1i = |g1|H (σ1)αW1 (ω1) = 1≤k

Proof. We continue with theorem of Burago and Ivanov, see the paper [BI12], page 635 (paragraph “proof of Theorem 2”). We present here a sketchy proof of this result for sake of completeness. We ﬁrst prove an auxiliary result from the plane geometry. 2 2 Proposition 7.2.1. Let K ⊂ R be a symmetric convex polygon, let f1, . . . , fn : R → R be linear functions such that fi K ≤ 1 for all i and let p1, . . . , pn be non–negative real P numbers such that i pi = 1. Then

X X 1 p p f ∧ f ≤ p p |f ∧ f | ≤ . i j i j i j i j H2(K) 1≤i

In addition, if K is a convex 2n–gon a1a2 . . . a2n, fi are supporting functions of K (that is, 2 2 fi = 1 on [ai, ai+1]) and pi = H (40aiai+1)/H (K) then the above inequalities turn into equalities. Proof. The proof splits into three elementary lemmata. 2 # Lemma 7.2.1. Let K = a1a2 . . . a2n be a symmetric 2n–gon in R . Let vi = aiai+1 for i = 1, . . . , n. Then

2 X X H (K) = |vi ∧ vj| = vi ∧ vj . 1≤i

Proof. The second identity follows from the fact that all pairs (vi, vj), 1 ≤ i < j ≤ n are of the same orientation. To prove the ﬁrst one, observe that n 2 2 X 2 H (K) = 2H (a1a2 . . . an+1) = 2 H (4a1ajaj+1), (7.7) j=2 because K is symmetric. Further

j 1 # # 1 X H2(4a a a ) = |a a ∧ a a | = |v ∧ v |, (7.8) 1 j j+1 2 1 j j j+1 2 i j i=1 # since ajaj+1 = v1 +v2 +. . . vj−1 and since all pairs (vi, vj), i < j are of the same orientation. The result now follows from identities (7.7) and (7.8).

2 # Lemma 7.2.2. Let K = a1a2 . . . a2n be a symmetric 2n–gon in R . Let vi = aiai+1 for 2 i = 1, . . . , n, fi : R → R a linear function such that fi = 1 on [ai, ai+1] and pi = 2 2 H (40aiai+1)/H (K). Then 1 p p |f ∧ f | = |v ∧ v |, i j i j H2(K)2 i j for all i, j, and therefore

X X 1 p p f ∧ f = |p p f ∧ f | = . i j i j i j i j H2(K) 1≤i

2 2 Proof. Denote Si = 2H (4), ai, ai+1) = |ai ∧ ai+1|, and hence pi = Si/H (K). Deﬁne a 2 ∗ 2 linear isometry (R ) → R as J(Sifi) = vi. Note that

SiSj|fi ∧ fj| = |Sifi ∧ Sjfj| = |vi ∧ vj|.

Therefore 1 1 p p |f ∧ f | = S S |f ∧ f | = |v ∧ v |. i j i j H2(K)2 i j i j H2(K)2 i j

To prove the second assertion, observe that all pairs (fi, fj), 1 ≤ i < j ≤ n are of the same orientation, hence

X X 1 X 1 p p f ∧ f = p p |f ∧ f | = |v ∧ v | = , i j i j i j i j H2(K) i j H2(K) 1≤i

2 Lemma 7.2.3. Let K = a1a2 . . . a2n be a symmetric 2n–gon in R . For each i = 1, . . . , n 2 let fi : R → R be a linear function such that fi = 1 on [ai, ai+1] and let pi be non–negative P real numbers satisfying pi = 1. Then X 1 p p |f ∧ f | ≤ . (7.9) i j i j H2(K) 1≤i

# 2 2 Proof. Denote vi = aiai+1, qi = 2H (40aiai+1)/H (K) and λi = pi/qi. By the previous lemma we have that 1 q q |f ∧ f | = |v ∧ v |. i j i j H2(K) i j 0 0 0 0 Let# v i= λivi for i from 1 to n. Consider a symmetric 2n–gon K = a1, . . . , a2n such that 0 0 0 ai, ai+1 = vi. Then X X |pipjfi ∧ fj| = |λiλjqiqj|fi ∧ fj| = 1≤i

0 0 Denote li = |vi| and li = |vi| = λi|vi|. Let hi denote the distance from the origin to the 2 2 line containing the side [ai, ai+1]. Then H (40, ai, ai+1) = hili hence qi = hili/H (K). Therefore X X 1 X 1 1 = p = λ q = λ h l = h l0. i i i H2(K) i i i H2(K) i i The last sum is the two-dimensional mixed volume V (K,K0), thus V (K,K0) = H2(K). By the Minkowski inequality we have that V (K,K0)2 ≥ H2(K)H2(K0). Therefore H2(K0) ≤ H2(K), and the lemma follows. 7.3 The existence for the Plateau problem. 87

To complete the proof of the proposition, it suffices to prove the previous lemma in a slightly more general setting, namely when K is a symmetric polygon (not necessary with 2n sides) and f1, . . . , fn are arbitrary linear functions such that fi K ≤ 1. This condition ∗ 2 ∗ means that fi belongs to the polar polygon K ⊂ (R ) . Consider the left hand side in the inequality (7.9) with others staying fixed. This function is convex (sum of absolute values of linear functions) therefore it attains its maximum at K∗ at a vertex of K∗. The vertices of K∗ are supporting linear functions to K at its sides. So it is sufficient to consider the case when each fi equals 1 on one of the sides of K. If two functions fi and fj coincide (with no loss of generality, f1 = fn) one reduces the problem to a smaller number of functions as follows: drop fn from the list of the functions and replace p1, p2, . . . , pn by p1 +pn, p2, . . . , pn−1. Also note that the changing of sign of one of the functions does not change the left–hand side of (7.9). Thus is suffices to consider the case when fi 6= fj and fi 6= −fj for all i 6= j. If n = 1 the left–hand side in (7.9) is zero so the inequality is trivial. If n > 1 we apply the previous lemma to the polygon 0 Sn K = i=1{x : |fi| ≤ 1} yields that X 1 1 p p |f ∧ f | ≤ ≤ , i j i j H2(K0) H2(K) 1≤i

Let us now derive the theorem from the proposition. Let σ = v1 ∧ v2 where v1, v2 are linearly independent vectors and consider a linear embedding I : R2 → X that takes the 2 −1 2 standard basis of R to v1 and v2. Let K = I (B), then φgBH (σ) = π/H (K). On the other hand, X α(σ) = π pkpl|fk ∧ fl|, 1≤k

7.3 The existence for the Plateau problem.

In this subsection we finish the proof of the existence result for the Plateau problem in case of two–dimensional rectifiable G chains. The following theorem is the main result of this chapter. Theorem 7.3.1. Let (X, || ... ||) be an n–dimensional Banach space and let G be a complete normed Abelian group. Then the mass M is lower semicontinuous with respect to flat convergence on R2(X,G). 88 The existence in the two dimensional case.

Proof. We start with a ﬁrst step where we prove the theorem in case when the limiting chain P is polyhedral.

Lemma 7.3.1. Let X and G be as in the formulation of the theorem. If Ti ∈ R2(X,G),P ∈ P2(X,G), and limi→∞ F(Ti − P ) = 0, then

(P ) ≤ lim inf (Ti). M i→∞ M Proof. Thanks to the weak approximation theorem in Banach spaces (see [Pau14], The- orem 4.4, page 329), we know that for every natural number i there exists a polyhedral chain Pi ∈ P2(X,G) such that F(Ti − Pi) ≤ 1/i and M(Pi) ≤ M(Ti) + 1/i. Next since F(Pi − P ) → 0, when i → +∞, we estimate with help of Corollary 7.2.1 1 M(P ) ≤ lim inf M(Pi) ≤ lim inf M(Ti) + = lim inf M(Ti), i→+∞ i→+∞ i i→+∞ and the lemma follows. Theorem 7.3.1 can now be proved with help of Lemma 7.3.1 and the strong approximation Theorem 5.2.1 exactly in the same way as Theorem 5.2.2.

As a consequence of our lower semicontinuity theorem, we obtain the following existence result. Theorem 7.3.2. Let (X, || ... ||) be an n–dimensional Banach space and let G be a locally compact complete normed Abelian group. Then the following Plateau problem admits a solution: ( minimize M(T ) T ∈ R2(X,G) such that ∂T = B, where B ∈ R1(X,G) has a compact support and satisﬁes ∂B = 0. Proof. We use the direct method of calculus of variations. Take a minimizing sequence, say Tj such that inf M(T ) = limj M(Tj). Let choose R0 such that B ⊂ B(0,R0). Let us prove that there exists a sequence of radii rk → ∞ such that rk ≤ rk+1 and that

c c lim sup (M(Tj B(0, rk) ) + M(∂(Tj B(0, rk) ))) = 0. k j

k c c c Denote Sj = Tj B(0, rk) and βk = supj (M(Tj B(0, rk) ) + M(∂(Tj B(0, rk) ))) . Note k that M(∂Sj ) = 0 once rk > R0, and hence it is left to prove that limk αk = 0 where k αk = supj M(Sj ). This can be proved exactly in the same way as in the paper [AS13] of Ambrosio and Schmidt. k Next, since for each j we have that spt(Sj ) ⊆ B(0, rk) using the diagonal argument, we infer that there exists a strictly increasing function φ : N → N such that for each k there k k k k+l holds F(Sφ(j) − S ) → 0. Note that S (B(0, rk)) = S (B(0, rk)). We are going to 7.3 The existence for the Plateau problem. 89 show that Sk converge towards some chain S. We start with showing that Sk is F–Cauchy. Indeed since the mass is lower semicontinuous

k k+l k k+l c M(S − S ) = M(S − S (B(0, rk)) ) ≤ k c k c lim inf (Sφ(j) (B(0, rk)) )) + lim inf (Sφ(j) (B(0, rk)) )) ≤ βk + βk+l. j M j M

Let us now show that Tφ(j) F–converges to S. Fix ε > 0 and ﬁnd K such that if k > K, then F(S (B(0, rk)) − S) ≤ ε. Hence

F(Tφ(j) − S) = F(Tφ(j) (B(0, rk)) − Tφ(j))+

F(Tφ(j) (B(0, rk)) − S (B(0, rk))) + F(S (B(0, rk)) − S) ≤ 2ε + βk, and the theorem follows. 90 The existence in the two dimensional case. Appendices

Appendix A

Lower semicontinuity and semiellipticity in the RNP spaces.

Throughout this appendix G denotes a locally compact normed Abelian group.

Deﬁnition A.0.1. Let X be an arbitrary Banach space. Call a function M : Pm(X,G) → R+ a mass if it satisﬁes •M (T + S) ≤ M(T ) + M(S)

•M (T + S) ≤ M(T ) + M(S), if Hm(set(T ) ∩ set(S)) = 0.

•M (T ) ≤ CM(T ), where M is the Hausdorﬀ mass and C depends only on X.

m •M (F#T ) ≤ Lip(F ) M(T ) for any piecewise linear mapping F : X → X and C depends only on X.

Remark A.0.1. Note that M(T ) is well deﬁned for any T ∈ Rm(X,G).

Deﬁnition A.0.2. Mass M satisﬁes triangle inequality if for any oriented simplexes σ, σ1, . . . σN and elements g, g1, . . . , gN ∈ G there holds

N N X X ∂(g[σ]) = ∂( gi[σi]) ⇒ M(g[σ]) ≤ M(gi[σi]). 1 1

Theorem A.0.3. Let X be a Banach space with Radon–Nykodym property. If mass M satisﬁes triangle inequality, then M is lower semi continuous on the space Rm(X,G) with respect to the ﬂat norm topology.

Remark A.0.2. This theorem gives a partial answer to a question, raised in the paper by Ambrosio and Kirchheim ([KA00] Appendix C).

Proof. The proof resembles that of Theorem 5.2.1. 94 Lower semicontinuity and semiellipticity in the RNP spaces.

Lemma A.0.2. Let X be an arbitrary Banach space. If mass M satisfies triangle inequality, then it is lower semi continuous on the space Pm(X,G) with respect to the F–topology. Proof. We start with a very special case when P = g[σ], i.e. when the chain P consists of only one simplex. Suppose that F(P − Pi) → 0 as i tends to ∞ for some Pi ∈ Pm(X,G). ∞ By the definition of flat convergence there exists a sequence of m–chains {Qi}i=1 and a ∞ sequence of (m + 1)–chains {Ri}i=1 such that P − Pi = Qi + ∂Ri and M(Qi) + M(Ri) → 0 when i tends to ∞. Note that ∂(P − Pi − Qi) = ∂(∂Ri) = 0, so we can apply the second condition to the chain (P − Pi − Qi) and obtain

M(P ) ≤ lim inf M(Pi + Qi) ≤ lim inf M(Pi) + C (Qi) ≤ lim inf (Pi), i i M i M and hence the ﬁrst case follows. We pass to the general case. Suppose that Pi → P in ﬂat norm for a sequence Pi ∈ Pm(X,G). For every j between 1 and N denote by Wj the m–plane containing the simplex

σj. Fix further a linear continuous projection PWj : X → Wj. We claim that there exist (m + 1)–convex polyhedrons Fj of the following form: Fj = conv{a1,j, . . . , am+1,j, vn,j, −vn,j} where

m+1 1 X v = a + ηω , n,j n k,i j k=1 for some (fixed) nonzero vector ωj ∈ Ker(PWj ) and a small number η chosen in a way that Fi and Fj do not overlap. Indeed, this follows from the finite dimensional case, Theorem 5.1.2 simply by regarding the linear subspace Y ⊂ X defined as Y = span{W1,...Wn ω1, . . . , ωN }. Next, let uj be a function defined as c c uj(x) = min{dist∞(PWj (x), σj ), dist∞(x, Fj )}, c where PWj is the fixed projection onto Wj, S stands for the complement of a set S and dist∞ is defined in the following way:

dist∞(x, S) = inf ||x − z||l∞(Y ), z∈S where l∞(Y ) is the supremum norm with respect to some Euclidean structure in Y . Note that it follows from the construction of the polyhedrons Fj that the sets {x : uj(x) > κ} are pairwise disjoint whenever κ > 0. The rest of the proof is almost identical to that of Theorem 5.1.2. We infer that for every j ∈ [1,...,N] and for every ε > 0 there exists δ > 0 such that for all r ∈ (0, δ) one has

M (gj[σj] − gj[σj ∩ {x : uj(x) > r}]) ≤ ε. (A.1)

We claim that for every j ∈ [1,...,N] there exists δe ∈ [δ/2, δ] and a sequence δi ∈ (δ/2, δ) such that δi → δe

F(P {x : uj(x) > δe} − Pi {x : uj(x) > δi}) → 0, (A.2) 95 when i tend to +∞. Indeed, by the deﬁnition of ﬂat norm, we know that for every εe > 0 there exists i = i(εe) such that for each i ≥ i(εe) there are Ri ∈ Pm+1(X,G) and Qi ∈ Pm(X,G) with M(Qi) + M(Ri) ≤ εe and P − Pi = Qi + ∂Ri. Hence for every ρ > 0 we can write

P {x : uj(x) > ρ} − Pi {x : uj(x) > ρ} = (Qi + ∂Ri) {x : uj(x) > ρ} =

Qi {x : uj(x) > ρ} + ∂(Ri {x : uj(x) > ρ})+

((∂Ri) {x : uj(x) > ρ} − ∂(Ri {x : uj(x) > ρ})) .

From here we see that for every i ≥ i(εe) and for every ρ > 0 one has

F(P {x : uj(x) > ρ} − Pi {x : uj(x) > ρ}) ≤ εe+ MhRi, uj, ρi. (A.3) On the other hand, referring to the slicing properties we infer that for every i there exists δi ∈ (δ/2, δ) such that MhRi, uj, δii ≤ 2M(Ri)/δ. Passing to a subsequence if necessary, we ∞ may assume that the sequence {δi}i=1 converges towards some δe. Note that

lim ||P ||({x : δe > uj(x) ≥ δi}) = 0. i→∞

Hence for the ﬁxed εe there exists number M(= M(εe)) such that ||P ||({x : δe > uj(x) ≥ δi}) ≤ εe once i > M. According to the inequality (A.3) applied to ρ = δi, for each i > M there holds

F(P {x : uj(x) > δe} − Pi {x : uj(x) > δi}) ≤

F(P {x : uj(x) > δi} − Pi {x : uj(x) > δi}) + F(P {x : δe > uj(x) ≥ δi}) ≤

2M(Ri) 2eε ε + + ||P ||({x : δe > uj(x) ≥ δi}) ≤ ε + + ε, e δ e δ e and the claim follows. We further derive an upper bound on the mass of the chains gj[σj] first using the inequality (A.1) then the first case of the proof and finally the fact that δi ≥ δ/2: δ M(gj[σj]) ≤ ε + M(P {x : uj(x) > δe}) ≤ ε + lim inf M(Pi {x : uj(x) > }). i→+∞ 2 We continue our estimates by taking the sum over all indexes j between 1 and N in the previous line:

N N X X δ M(P ) = M(gj[σj]) ≤ Nε + lim inf M(Pi {x : uj(x) > }) ≤ Nε + lim inf M(Pi), i→+∞ 2 i→+∞ j=1 j=1 where the last inequality follows from the fact that the sets {x : uj(x) > δ/2} are pairwise disjoint. The ﬁrst implication will now follow after we let ε tend to 0. 96 Lower semicontinuity and semiellipticity in the RNP spaces.

Lemma A.0.3. Let X be an arbitrary Banach space and let M ⊂ X be a C1–smooth manifold of dimension m with a ∈ M. Fix t > 0. There exists r0 such that for all r < r0 there is a Lipschitz mapping F : X → X such that

• F (x) = x once x∈ / U(a, r) • F (M ∩ B(a, tr)) ⊆ W where W = Tan(a, M)

• Lip(F ), Lip(F −1) ≤ 1/t.

Proof. Suppose as we can that a = 0. Choose, according to the Hahn–Banach theorem, a linear continuous projection P = PW : X → W such that ||P || ≤ C(m). Note that there exists ρ > 0 such that M ∩ B(0, ρ) ⊆ ψ(W ∩ B(0, ρ)), where we write ψ := (P (M ∩ U(0, r)))−1 : P (M ∩ U(0, r)) → M ∩ U(0, r) for brevity. Take ε = ε(t) to be deﬁned by the end of the proof and choose r0 ≤ ρ that guarantees |||Dψ(ξ) − idW ∩B(0,r0)||| ≤ ε for all ξ ∈ B(0, r0). Write Taylor’s formula for ψ: for z1, z2 ∈ W ∩ B(0, r) there holds

ψ(z1) = ψ(z2) + Dψ(z2)(z1 − z2) +o ¯(||z1 − z2||) =

ψ(z2) + (z1 − z2) + (Dψ(z2) − idW ∩B(0,r0))(z1 − z2) +o ¯(||z1 − z2||).

Hence for some r1, for all r < r1 and z1, z2 ∈ W ∩ B(0, r) one has

||z1 − ψ(z1) − z2 + ψ(z2)|| ≤ ε||z1 − z2||.

Using almost the same argumentation, one obtains that there exists r2 such that for all z ∈ B(0, r2) ||ψ(z) − z|| ≤ ε||z||.

Deﬁne further for r < r0 and x ∈ X a function u : X → [0, 1] as

||x|| u(x) = γ , r where γ is the piecewise linear function satisfying γ(z) = 1 for z ∈ [0, t] and γ(1) = 0. We are now ready to deﬁne the desired function F . Let F (x) = x for x∈ / U(0, r) and let F (x) = x + u(x) · [P (x) − ψ(P (x))] otherwise. Let us check the assertions of the lemma. The ﬁrst and the third are clear, let us estimate the Lispchitz constants of F and F −1. If x, y ∈ B(0, r) then we estimate using the inequalities that we have derived from Taylor’s formula:

||F (x) − F (y)|| ≤ ||x − y|| + ||u(x)(P (x) − ψ(P (x))) + u(y)(ψ(P (y)) − P (y))|| ≤ ||x − y|| + |u(x) − u(y)| · ||P (y) − ψ(P (y))|| + ||P (x) − ψ(P (x)) + ψ(P (y)) − P (y)|| ≤ ||x − y|| 1 ≤ ||x − y|| + ε||P (y)|| + ε||P (x) − P (y)|| ≤ ||x − y||, 1 − t t 97 with the last inequality valid for ε small enough. If y ∈ B(0, r) and x∈ / B(0, r) then we estimate in the following way:

||F (x) − F (y)|| ≤ ||x − y|| + |u(y)| · ||P (y) − ψ(P (y))|| ≤ 1 − ||y|| C(m)ε 1 ||x − y|| + r ε||P (y)|| ≤ ||x − y|| 1 + ≤ ||x − y||, 1 − t r(1 − t) t where the last inequality holds for ε small enough. The function F −1 can be treated analogously. Theorem A.0.4. Let X be a Banach space with the Radon Nykodim property and let G be a complete normed Abelian group. If T ∈ Rm(X,G) with 1 ≤ m ≤ n then for every ε > 0 there exists a polyhedral chain P ∈ Pm(X,G) and a Lipschitz mapping F : X → X such that • ||F (x) − x|| ≤ ε for all x ∈ X and F (x) = x once dist(x, set(T )) ≥ ε;

• max{Lip(F ), Lip(F −1)} ≤ 1+ε, where the Lipschitz constants are taken with respect to the norm || ... ||;

• M(F#T − P ) ≤ ε. Proof. Let us derive the approximation theorem A.0.4 from the previous lemma. Since m T ∈ Rm(X,G), we know that T = θ(x)H M, for some m–rectiﬁable set M which we will call set(T ) and a Borel measurable function θ : set(T ) → G such that θ ∈ L1(X,G) (which means that the function |θ| ∈ L1(Hm set(T ))). Since set(T ) is m–rectiﬁable and since X has Radon Nykodim property, there exist countably many C1–smooth manifolds ∞ m {Mi}i=1 and a set M0 such that H (M0) = 0 satisfying [ set(T ) ⊆ Mi ∪ M0. i≥1

S m of the previous lemma. Since set(T ) = j (Mj ∩ B(aj, rj)) ∪ N with H (N) = 0, we can ﬁnd a number N = N(ε) such that

∞ ! X M Tj ≤ ε, (A.5) j=N+1 where we denote Tj = T (Mj ∩ B(aj, rj)) for sake of brevity. The next step consists in proving that for every δ > 0 there exists a polyhedral chain Ψj ∈ Pm(X,G) such that M(Fj#Tj − Ψj) ≤ δ. (A.6) −1 m First of all, we know that ||Fj#Tj|| = (Fj ◦ θ)H Fj(Mj ∩ B(aj, rj)). Since the function θ is integrable on the set Mj ∩ B(aj, rj), according to the Lusin theorem for each δ > 0 we −1 can ﬁnd a compact subset C ⊂ Fj(Mj ∩ B(aj, rj)) such that (Fj ◦ θ) C is continuous and that ||Fj#Tj||(Fj(Mj ∩B(aj, rj))\C) ≤ δ. Suppose, as we can that C ⊆ {x : θ(x) 6= 0}. Second of all, we decompose C into a ﬁnite union of sets in a way that

K [ C = Ci ∪ C0 i=1 with ||Fj#Tj||(C0) ≤ δ and osc(θ Ci) ≤ δ infC |θ|. Third of all, for each natural i between 1 and K there exists Si ⊂ Fj(Mj ∩ B(aj, rj)), a finite union of dyadic m–semi–cubes such m m that H (Ci4Si) ≤ δH (Ci). Finally, for each i ∈ [1,...,K] we choose an element θi ∈ G that guarantees |θi − θ(x)| ≤ δ infCi |θ| for each x ∈ Ci. We define the desired polyhedral chain as K X Ψj = θi[Si]. i=1 Let us estimate the mass of the following difference

Z K m X m M(Fj#Tj − Ψj) ≤ |θ(x)|dH (x) − |θi|H (Si) ≤ i=1 Fj (Mj ∩B(aj ,rj )) K Z K X m X m δ + |θ(x)|dH (x) − |θi|H (Si) ≤ i=1 i=1 Ci K X m m δ + δ||Fj#Tj||(C) + |θi| (H (Ci) − H (Si)) ≤ δ + δ(2 + δ)||Fj#Tj||(C), i=1 and the claim (A.6) follows. Now we are ready to ﬁnish the proof of the theorem. We choose

N X F := F1 ◦ ... ◦ FN and P := Ψj. j=1 99

By the construction we see that F and P satisfy the ﬁrst two claims of the theorem. Let us check the third one:

∞ N X X M(F#T − P ) = M(F# T B(aj, rj) − P ) ≤ M( Fj#T B(aj, rj) − P ) + ε ≤ ..., j=1 j=1 where the last inequality follows from the line (A.5). We continue the estimate, ﬁrst using the estimate (A.4) and afterwards the inequality (A.6) with δ = ε/N :

N N ! N X X X Nε ... ≤ F T − Ψ + ε ||T || (B(a , r )) + ε ≤ + ε + ε (T ). M j# j j j j N M j=1 j=1 j=1

Since ε > 0 is arbitrary, the theorem follows.

Let us now prove our lower semicontinuity result. Let Tj be a sequence of m–rectifiable G chains, converging in flat norm to some T ∈ Rm(X,G). Fix ε > 0. According to the storng approximation theorem, there exist a P ∈ Pm(X,G) and F : X → X such that P = F#T +E with M(E) ≤ ε. Note that since F is Lipschitz, one has limj→∞ F(F#Tj −F#T ) = 0 and hence also limj→∞ F(E + F#Tj − P ) = 0 by the definitions of E and P . Since P is polyhedral, we can use the lemma:

M(E + F#T ) = M(P ) ≤ lim inf M(E + F#Tj) ≤ C (E) + lim inf M(F#Tj) j→∞ M j→∞ m m Cε + lim inf(Lip(F )) M(Tj) ≤ Cε + (1 + ε) lim inf M(Tj), (A.7) j→∞ j→∞ where the last inequality follows from the fact that F is (1 + ε)–Lipschitz. On the other hand, since Lip(F −1) ≤ 1 + ε, we infer that

−m M(E + F#T ) ≥ M(F#T ) − M(E) ≥ (1 + ε) M(T ) − Cε. (A.8)

The lines (A.7) and (A.8) now give

−m m (1 + ε) M(T ) ≤ (1 + ε) lim inf M(Tj) + 2Cε, j→∞ and the theorem will follow after we let ε tend to zero. 100 Lower semicontinuity and semiellipticity in the RNP spaces. Index

B(spt(T ), ε), 30 φBH , 33 , 29 , 61 σB, 34 φgBH , 82 Hom(X,X), 69

B.Kirchheim’s theorem, 30 Besicovitch covering theorem, 31 Burago–Ivanov’s theorem, 82 Busemann–Hausdorﬀ density, 33 constancy theorem, 30

Densities of measures, 47 Dyadic semi–cubes, 26

Good choice of good projections, 71 Good projections, 47 Gross mass, 73 Gross measure, 71

Multiplicity function, 74

Polish space, 70

Strong approximation theorem, 61 Supporting hyperplane, 25 Suslin set, 71

Tangent measures, 47 Top–dimensional chains, 28

Universally measurable function, 75 Universally measurable set, 74

Weak approximation theorem, 30 102 INDEX Bibliography

[Bus49] Herbert Busemann. “A theorem on convex bodies of the Brunn-Minkowski type”. In: Proceedings of the National Academy of Sciences USA 35 (1949), pp. 27–31. [Bus50] Herbert Busemann. “The Foundations of Minkowskian Geometry”. In: Commen- tarii mathematici Helvetici 24 (1950), pp. 156–187. [BS60] Herbert Busemann and Ernest Strauss. “Area and normality”. In: Pacific Journal of Mathematics 10 (1960), pp. 35–72. [Fed69] Herbert Federer. Geometric Measure Theory. Springer–Verlag, 1969. [BZ88] Yuri Burago and Victor Zalgaller. Geometric Inequalities. Springer Verlag, 1988. [Kir94] Bernd Kirchheim. “Rectifiable metric spaces: local structure and regularity of the Hausdorff measure”. In: Procedings of the Americal Mathematical Society 121 (1994), pp. 113–123. [Tho96] Anthony Thompson. Minkowski Geometry. Cambridge University Press, 1996. [Mat99] Pertti Mattila. Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge University Press, 1999. [KA00] Bernd Kirchheim and Luigi Ambrosio. “Currents in metric spaces”. In: Acta Mathematica 185 (2000), pp. 1–80. [Sch01] Rolf Schneider. “On the Busemann Area in Minkowski Spaces”. In: Beitrage zur Algebra und Geometrie Contributions to Algebra and Geometry 42 (2001), pp. 263–273. [TA04] Anthony Thompson and Juan Alvarez. “Volumes on normed and Finsler spaces”. In: journal volume (2004), pages. [Lel08] Camillo De Lellis. Rectifiable Sets, Densities and Tangent Measures. Zurich Lec- tures in Advanced Mathematics, 2008. [BI12] Dmitri Burago and Sergei Ivanov. “Minimality of planes in normed spaces”. In: Geometric and Functional Analysis 22 (2012), pages. [PH12] Thierry De Pauw and Robert Hardt. “Rectifiable and flat G chains in a metric space”. In: American Journal of Mathematics 134 (2012), pp. 1–69. 104 BIBLIOGRAPHY

[AS13] Luigi Ambrosio and Thomas Schmidt. “Compactness results for normal currents and the Plateau problem in dual Banach spaces”. In: Procedings of the London Mathematical Society 106 (2013), pp. 1121–1142. [Coh13] Donald Cohn. Measure Theory: Second Edition. Birkhauser, 2013. [Pau14] Thierry De Pauw. “Approximation by Polyhedral G Chains in Banach Spaces”. In: Zeitschrift Analysis und ihre Anwendungen 33 (2014), pp. 311–334. [PH14] Thierry De Pauw and Robert Hardt. “Some basic theorems on ﬂat G chains”. In: Journal of Mathematical Analysis and Applications 418 (2014), pp. 311–334. [BP15] Philippe Bouaﬁa and Thierry De Pauw. “Integral geometric measure in separable Banach space”. In: Mathematische Annalen 363 (2015), pp. 311–334. [EG15] Lawrence Evans and Ronald Gariepy. Measure Theory and Fine Properties of Functions. Cambridge University Press, 2015.