SANTALO´ POINT FOR THE HOLMES-THOMPSON BOUNDARY AREA

FLORENT BALACHEFF, GIL SOLANES, AND KROUM TZANEV

Abstract. We explore the notion of Santal´opoint for the Holmes-Thompson boundary area of a convex body in a normed space. In the case where the is C1, we prove existence and uniqueness. When the normed space has a smooth quadratically convex unit ball, we exhibit a dual Santal´opoint expressed as an average of centroids of projections of its dual body.

1. introduction

Several decades ago Santal´ostudied in [Santal´o1949] the functional x int(K) (K x)◦ (1.1) ∈ 7→ | − | n n ◦ associated to any convex body K of R . Here denotes the Lebesgue measure of R , and A = n n | · | y R x, y = i=1 xiyi 1 x A is the polar set of a convex body A. He found that this {functional∈ | h is properi and strictly≤ convex.∀ ∈ Consequently,} there exists a unique minimizing point s(K) in the interior of KP, known nowadays as the Santal´opoint of K. Along the way, Santal´ocomputed the derivative of (1.1) and showed that d (K tv)◦ = (n + 1) c(K◦), v (1.2) dt | − | h i t=0 ◦ ◦ n where c(K ) = K◦ x dx is the centroid of the polar body K and v any vector in R . Therefore s(K) lies at the origin if and only if c(K◦) lies at the origin. Determining the Santal´opoint of a given convex body is aR very difficult question, so this characterization appears to be particularly useful. n As the Lebesgue measure coincides with the n-dimensional Hausdorff measure of R , it is natural to consider the following functional similar to (1.1) x int(K) H n−1(∂(K x)◦), (1.3) ∈ 7→ − where H n−1 denotes the (n 1)-dimensional Hausdorff measure, and look for possible minimizing points. − More generally, let V be a n-dimensional real . To any convex body B in V that contains the origin in its interior is associated a unique (asymmetric) norm B whose unit ball is precisely k · k B. The restriction of B to any hypersurface defines a Finsler metric whose corresponding Holmes- k·k Thompson (n 1)-volume (or area in short) will be denoted by AB( ). If K is a convex body in − ∗ ∗ · (V, B), we can consider its dual body K = p V p(x) 1 x K as a convex body of k · k ∗ { ∈ | ≤ ∀ ∈ } the dual (V , B∗ ) and focus on the associated Holmes-Thompson area of its k · k ∗ n arXiv:2012.11440v1 [math.MG] 21 Dec 2020 boundary sphere, that is the quantity AB∗ (∂K ). In the special case where V = R and B is the ∗ n−1 ◦ Euclidean unit ball, we have AB∗ (∂K ) = H (∂K ). Therefore functional (1.3) appears to be a particular case of the functional ∗ x int(K) AB∗ (∂(K x) ). ∈ 7→ − A minimizing point of this functional will be called a Santal´opoint of K for the Holmes-Thompson area in (V, B). k · k 2020 Mathematics Subject Classification. Primary: 52A20, 52A40, 53C65. Secondary: 52A38. Key words and phrases. Convex body, Crofton formula, Hausdorff measure, Holmes-Thompson area and volume, Minkowski geometry, Santal´opoint, symplectic geometry. The first author acknowledges support by the FSE/AEI/MICINN grant RYC-2016-19334 “Local and global systolic geometry and topology”. The second author is supported by the Serra Hunter Programme. The first and the second author acknowledge support by the FEDER/AEI/MICIU grant PGC2018-095998-B-I00 “Local and global invariants in geometry”. 1 2 FLORENT BALACHEFF, GIL SOLANES, AND KROUM TZANEV

In this paper we first prove existence and uniqueness of Santal´opoints for the Holmes-Thompson area in any normed vector space whose unit ball is of class C1.

Theorem 1.1. Let K be a convex body in a finite-dimensional real normed vector space (V, B) whose unit ball B is of class C1. The functional k · k ∗ x int(K) AB∗ (∂(K x) ) ∈ 7→ − is strictly convex and proper on the interior of K.

In particular there exists a unique minimizing point B(K) int(K) for this functional. S ∈ In case the unit ball is no longer of class C1, the above functional is still convex (and proper) on the interior of K, but strict convexity is no longer guaranteed. Therefore, Santal´opoints do exist but are no longer necessarily unique. In section3, our study goes far beyond the C1-regularity, and we give precise necessary and sufficient conditions for the uniqueness. For example, when the unit ball B is a polytope there is always a convex body K with an infinite set of Santal´opoints, as illustrated in a concrete example 3.6.

Note that Theorem 1.1 yields in particular a map K K (K) well defined for each convex body K with C1 boundary. By invariance properties of the→ Holmes-ThompsonS area, this map is affinely equivariant in the sense that TK (TK) = T ( K (K)) S S for any invertible affine map T . See Remark 3.2 for more details.

In order to characterize B(K), we study the first variation of the above functional. Assuming S that the normed vector space (V, B) is Minkowski, i.e. B is smooth and quadratically convex, we obtain an interesting formula closelyk · k ressembling (1.2) and involving some classical notions from affine differential geometry.

Given x ∂B, recall that the affine normal line is a canonical 1-dimensional linear subspace Nx V ∈ ⊂ transverse to Tx = Tx∂B, which behaves equivariantly under linear transformations of ∂B. We thus have the decompositions ∗ ⊥ ⊥ V = Tx Nx and V = N T . ⊕ x ⊕ x ⊥ ∗ ∗ ⊥ Let us consider the isomorphism Nx Tx given by φ φ Tx , and the projection πx : V Nx ⊥ ' 7→ | → with kernel ker πx = Tx . Pick an arbitrary smooth measure µ on ∂B and consider the Lebesgue ∗ ∗ ⊥ ∗ measure µx on Tx, and its dual measure (µx) on Tx . Let νx be the Lebesgue measure on Nx Tx ∗ ' corresponding to (µx) .

Definition 1.2. Let K be a convex body in a Minkowski space (V, B). Using the previous notation, we define k · k

∗ B(K ) = q dνx(q) dµ(x). C ∗ Z∂B Zπx(K ) !

∗ Note that the inner integral is the centroid of the projection πx(K ) with respect to νx and belongs ∗ ∗ to V . It is easy to see that B(K ) does not depend on the choice of µ. C Our second main result is the following.

Theorem 1.3. Let K be a convex body in a Minkowski space (V, B). Then for any v V we have k · k ∈ d ∗ n + 1 ∗ AB∗ (∂(K tv) ) = B(K ), v . (1.4) dt − εn−1 hC i t=0 ∗ ∗ In particular B(K) lies at the origin of V if and only if B(K ) lies at the origin of V . S C Here εn−1 denotes the Euclidean volume of a (n 1)-dimensional unit Euclidean ball and , the canonical pairing. Note the similarity with− formula (1.2). h· ·i In the particular case of a convex body in an Euclidean space, both theorems sum up into the following. HOLMES-THOMPSON AREA & SANTALO´ POINT 3

n Corollary 1.4. Given a convex body K in R , there exists a unique point (K) int(K) minimizing the functional S ∈ x int(K) H n−1(∂(K x)◦). ∈ 7→ − Furthermore, this point lies at the origin if and only if the following average of centroids of orthogonal projections of its polar body

C (K◦) = y dλ(y) dσ(x) n−1 π (K◦) ZS Z x⊥ ! lies at the origin.

Here πx⊥ denotes the orthognal projection onto the hyperplane orthogonal to x, and σ, λ denote n−1 ⊥ respectively the canonical measure on S and the Lebesgue measure on x induced by the standard Euclidean structure. This paper is organized as follows. In section2 we collect some material on Holmes-Thompson volume, and present a Crofton formula for the Holmes-Thompson area of a hypersurface in a Minkowski space due to [Alvarez´ Paiva 1998]. This formula will be decisive to show (1.4). In section3 we prove Theorem 1.1. The next three sections are devoted to the proof of Theorem 1.3. The strategy consists in using the above mentioned Crofton formula which describes the Holmes-Thompson area of ∂K∗ as an integral on the set of lines intersecting K∗. In order to compute its first variation under translations, we first use the boundary sphere ∂B to construct a parametrization space for the oriented affine lines of the dual vector space in terms of intersections of affine hyperplanes. We then show how to rewrite the measure involved in Alvarez´ Paiva’s formula in this new space. The effect of translations is then simple to describe which allows us to compute the first variation of the area. So after shortly recalling in section4 some classical notions of equiaffine differential geometry, we define the parametrization space in section5 and show how to rewrite the measure of lines in this new setup. In the last section6 we compute the first variation of the Holmes-Thompson area and obtain formula (1.4).

2. Background on the Holmes-Thompson notions of volume and area

In this section we collect some material on Holmes-Thompson notion of volume and area. The two major tools we will use in the sequel are a Crofton type formula due to Alvarez´ Paiva and presented in Proposition 2.1, and a duality formula due to Holmes & Thompson stated in Proposition 2.3.

2.1. Holmes-Thompson volume on continuous Finsler manifolds. Let M be a smooth man- ifold M of dimension k endowed with a continuous Finsler metric F : TM [0, ), that is a continuous function which is positively homogenous of degree 1 outside the zero→ section∞ and such that for all x M the subsets of TxM ∈ Dx(M,F ) := v TxM F (x, v) 1 { ∈ | ≤ } are convex bodies containing the origin in their interior (or equivalently, F is a continuous function whose restriction to each tangent space TxM is a norm, possibly asymmetric). For any measurable set A M denote by D∗(A, F ) the unit co-disc bundle over A consisting of the disjoint union of the ⊂ ∗ ∗ dual bodies Dx(M,F ) (TxM) for x A. The Holmes-Thompson volume of the measurable set A in (M,F ) is then defined⊂ as the following∈ integral:

1 k (A, F ) = εk ωM V k! ∗ ZD (A,F ) ∗ where ωM denotes the standard symplectic form on the cotangent bundle T M. If ρ denotes any non-vanishing k-density on M, then ∗ ∗ Dx(M,F ) ρx (A, F ) = | | ρx V εk ZA ∗ ∗ where is the dual Lebesgue measure on T M of the Lebesgue measure ρ on TxM associated | · |ρx x | · | x to ρ. More precisely, ρ is the Lebesgue measure on TxM normalized to give volume 1 on a | · | x 4 FLORENT BALACHEFF, GIL SOLANES, AND KROUM TZANEV

∗ parallelotope spanned by a base v1, . . . , vk of TxM for which ρx(v1, . . . , vk) = 1, and ρx is the ∗ | · | Lebesgue measure on Tx M normalized to give volume 1 for the parallelotope spanned by the covectors ∗ ξ1, . . . , ξk T M forming a base which is dual to (v1, . . . , vk). ∈ x 2.2. Holmes-Thompson volume and area in normed spaces. Let B be a convex body in a n-dimensional real vector space V that contains the origin as an interior point. The associated norm B defines a continuous Finsler metric on the manifold V whose Holmes-Thompson volume satisfies fork·k any compact set A the equality A B∗ ∗ (A, B) = | | | | V k · k εn where and ∗ are respectively the measure and its dual measure associated to some translation- | · | | · | invariant density on V . We will denote shortly (A, B) by B(A). V k · k V n ◦ In the case where V = R we find that B(A) = A B /εn. In particular the Finsler volume B(B) V | | | | V of the convex body B itself coincides with its volume product up to the constant 1/εn.

It turns out that the associated norm B also permits to define for any hypersurface M V (and, k · k ⊂ more generally, for any smooth submanifold of V ) a continuous Finsler metric FB : TM [0, ) by → ∞ simply setting FB(x, v) = v B. We will denote shortly (M,FB) by AB(M) and speak of Holmes- Thompson area of M. Thisk k notion of area extends to boundariesV of convex bodies. For this recall that the set of singular boundary points (those for which the supporting hyperplane is not uniquely defined) of a convex body K in V is of (n 1)-dimensional Hausdorff measure zero, see [Reidemeister 1921]. Therefore the formula − ∗ ∗ (TxM B) ρ AB(M) = | ∩ | ρ(x) (2.1) ε ZM n−1 where ρ denotes any translation-invariant (n 1)-density on V makes sense when M = ∂K is the − boundary of a convex body K. Moreover, if Kn K is a sequence of smooth convex bodies that → converges in the Hausdorff metric to a convex body K, then AB(∂Kn) AB(∂K) (see Remark 2.2). →

2.3. Alvarez´ Paiva formula for hypersurfaces in Minkowski spaces. In the case the normed space (V, B) has smoothness and strict convex properties, the Holmes-Thompson area of hyper- surfaces admitsk · k a Crofton formula we present now. Let B be a smooth quadratically convex body of V that contains the origin as an interior point. Then the associated normed space (V, B) is said to be Minkowski. In that case its dual body is also smooth and quadratically convex,k and · k the dual Legendre transform ∗ : ∂B∗ ∂B L → is a well defined diffeomorphism. Recall that ∗(p) is defined as the unique x ∂B such that L ∈ p(x) = 1. Denote by G+(V ) the space of oriented affine lines in V whose elements are denoted by x + v + where x is a point in V and v + is the 1-dimensional vectorial subspace of V spanned and h i h i oriented by a vector v. In particular according to this notation x + v + = y + w + if and only if h i h i there exists (t, s) R R>0 such that y = x + tv and w = sv. Now consider the projection map ∈ × ∗ π : V ∂B G+(V ) × → ∗ (x, p) x + (p) +. 7→ hL i ∗ ∗ ∗ Recall that ωV denotes the standard symplectic form of T V and denote by i : V ∂B , T V the × → canonical inclusion. It is well known that there exists a unique symplectic form ωB on G+(V ) such ∗ ∗ that π ωB = i ωV , see [Arnold & Givental 1990, Besse 1978]. Here is the Crofton formula associated to this symplectic form. Proposition 2.1 (Alvarez´ Paiva). The Holmes-Thompson area of a compact immersed hypersurface M in a Minkowski space (V, B) satisfies the following formula: k · k 1 n−1 AB(M) = #(L M) ωB . 2 (n 1)! εn−1 ∩ − ZG+(V ) HOLMES-THOMPSON AREA & SANTALO´ POINT 5

Applying this formula to the boundary of a smooth convex K in V we obtain the following formula:

1 n−1 AB(∂K) = ωB . (n 1)! εn−1 − ZL∩K6=∅ By continuity we see that this formula still holds when the convex body K is not necessarily smooth. Proposition 2.1 was first stated in [Alvarez´ Paiva 1998] (see [Alvarez´ & Fernandes 1998, Theorem 3.1]) more generally for immersed hypersurfaces in reversible Finsler manifolds whose space of oriented geodesics is a smooth manifold. We include the proof here for the reader’s convenience as well to check that it is still true in our context without the reversibility assumption.

∗ ∗ ∗ ∗ Proof. First note that if H V is a linear subspace, then (B H) = rH∗ (B ) where rH∗ : V H ⊂ ∩ → denotes the restriction morphism defined by rH∗ (p) = p|H . Thus ∗ ∗ ∗ D (M,FB) = (x, p) T M x M and p (B TxM) { ∈ ∗ | ∈ ∈ ∩ ∗ } ∗ = (x, p) T M x M and p r(TxM) (B ) { ∈ ∗ | ∈ ∈ ∗} ∗ = (x, p) T M x M and p r(TxM) (∂B ) { ∈ ∗ | ∈ ∈ } = PM (M ∂B ) × where ∗ ∗ PM : M ∂B D (M,FB) × → (x, p) (x, p ). 7→ |TxM Therefore n−1 1 ∗ n−1 (n 1)! εn−1 AB(M) = ωM = PM ωM . − ∗ 2 ∗ ZD (M,FB ) ZM×∂B ∗ ∗ Now observe that PM ωM = i ωV (as both are the exterior derivative of the tautological one-form) ∗ ∗ which implies using the identity i ωV = π ωB that

∗ n−1 n−1 2 (n 1)! εn−1 AB(M) = π ωB = #(L M)ωB . − ∗ ∩ ZM×∂B ZG+(V ) 

Observe that we obtain by the way the following formula for Holmes-Thompson area:

1 ∗ n−1 AB(∂K) = i ωV . (2.2) 2 (n 1)! εn−1 ∗ − Z∂K×∂B Remark 2.2. We easily check from the above formula that if Kn K is a sequence of smooth convex → bodies that converges in the Hausdorff metric to a convex body K, then AB(∂Kn) AB(∂K). → 2.4. Holmes-Thompson duality formula. To conclude this section let us recall the following duality principle for the Holmes-Thompson area of convex bodies. Proposition 2.3 (Holmes & Thompson). Let K and B be two convex bodies in V that contain the origin in their interior. Then ∗ AB∗ (∂K ) = AK (∂B).

Proof. This was first observed by Holmes and Thompson in [Holmes & Thompson 1979]. Here is a quick argument:

1 n−1 AK (∂B) = ωV 2 (n 1)! εn−1 ∗ − Z∂B×∂K 1 n−1 = ωV ∗ 2 (n 1)! εn−1 ∂K∗×∂(B∗)∗ − ∗ Z = AB∗ (∂K ), ∗ where we have abusively denoted i ωV by ωV .  6 FLORENT BALACHEFF, GIL SOLANES, AND KROUM TZANEV

3. Strict convexity and properness

In this section we prove Theorem 1.1 which we restate as follows using Proposition 2.3. Theorem 3.1. Let B and K be two convex bodies of a finite-dimensional real vector space V . Suppose that ∂B is of class C1. The functional

x AK−x(∂B) 7→ is strictly convex and proper on the interior of K.

In particular there exists a unique minimizing point B(K) int(K) for the Holmes-Thompson area ∗ ∗ S ∈ of the boundary sphere of (K x) in (V , B∗ ). − k · k We split the proof into two subsections. The first concerns the strict convexity of the functional above. The second proves the properness of the functional, using an isoperimetric inequality between Holmes-Thompson notions of volume and area. Remark 3.2. Observe that in Theorem 3.1 we no longer need B to contain the origin as an interior point. Therefore the Santal´opoint K (K) int(K) is well defined for every convex body K with C1 boundary. Now using (2.2) we easilyS see∈ that

ALK (∂LB) = AK (∂B) for every invertible L: V V . So, if T : V V is an invertible affine map, we get that → → AK−x(∂B) = ATK−T (x)(∂ T B) for all x int(K). The Santal´opoint thus fulfills ∈ TB(TK) = T ( B(K)). S S In particular the map K K (K) is affinely equivariant. → S n 3.1. Proof of the strict convexity. After fixing some isomorphism V R , we can use the standard Euclidean structure to identify the dual body A∗ with the polar body' A◦ of any convex body A containing the origin as an interior point, and rewrite ◦ (Ty∂B (K x)) n−1 AK−x(∂B) = | ∩ − | dλn−1(y) (3.1) ε Z∂B n−1 using formula (2.1). Here we have denoted by

dλn−1 the (n 1)-dimensional volume element induced on hypersufaces by the Euclidean • structure − C n−1 = C dλn−1 the (n 1)-Hausdorff measure of any compact domain C contained in •| some| hyperplane. − R Recall that a point y in the boundary of a convex body B is said to be regular if there is a unique supporting hyperplane y + H(y, B) of B at y. The set reg B ∂B of regular points has (n 1)- ⊂ − Hausdorff measure H n−1(reg B) = H n−1(∂B) (see [Reidemeister 1921]), and the map n y reg B H(y, B) Grn−1(R ) ∈ 7→ ∈ n is continuous (cf. [Schneider 2013, Lemma 2.2.12]). Here Grk(R ) denotes the set of all vector k- n n−1 planes in R . Hence, the push forward of the (n 1)-Hausdorff measure H by this map is well − n defined and we denote by µ∂B this measure on Grn−1(R ). The measure µ∂B coincides with the push n−1 forward of the classical surface area measure of B (which is a measure on the sphere S ) using the n−1 n standard two folding application S Grn−1(R ). → Consider the functional n fK : Grn−1(R ) int(K) (0, ) × → ∞ ◦ (H, x) (H (K x)) n−1 7→ | ∩ − | which is continuous as a composition of continuous applications. HOLMES-THOMPSON AREA & SANTALO´ POINT 7

n Lemma 3.3. For any H Grn−1(R ), the function x fK (H, x) is convex on int(K). ∈ 7→ Proof. By [Schneider 2013, eqs. (1.52) & (1.53)], in a m-dimensional Euclidean vector space (W, , ), the Lebesgue measure of any convex body C can be computed using the formula h· ·i 1 1 m C m = dλm−1(u), | | m m−1 h ◦ (u) ZS (W )  C  m−1 where S (W ) denotes the unit sphere in (W, , ) and hC◦ is the support function of the polar ◦ h· ·i ◦ body C , defined for any x W by hC◦ (x) = max x, y y C . Therefore ∈ {h i | ∈ } 1 1 n−1 fK (H, x) = dλn−2(u). n 1 n−2 h (u) − ZS (H)  H∩(K−x)  n−2 Now observe that u S (H) the function x hH∩(K−x)(u) is concave as for any λ (0, 1) and ∀ ∈ 7→ ∈ x1, x2 int(K) we have ∈ h (u) = max u, z z H [λ(K x1) + (1 λ)(K x2)] H∩(K−λx1−(1−λ)x2) {h i | ∈ ∩ − − − } max u, z z λ[H (K x1)] + (1 λ)[H (K x2))] ≥ {h i | ∈ ∩ − − ∩ − } = λh (u) + (1 λ)h (u). H∩(K−x1) − H∩(K−x2) The function t 1/tn−1 being strictly convex on (0, ), we get that the function 7→ ∞ 1 n−1 x 7→ h (u)  H∩(K−x)  is convex. So fK (H, ) is also convex as an integral of convex functions. ·  n For any x1 = x2 int(K) let us define StrcvxK (x1, x2) Grn−1(R ) as the set of hyperplanes H such that 6 ∈ ⊂ fK (H, x1) + fK (H, x2) 2 fK (H, (x1 + x2)/2) > 0. (3.2) − Observe that StrcvxK (x1, x2) is an open set by continuity of fK . Furthermore, for a fixed H, we know by [Santal´o1949] that for any x0 int(K) the map x fK (H, x + x0) is strictly convex ∈ 7→ on H int(K x0). So in particular, we always have StrcvxK (x1, x2) = as it contains any ∩ − 6 ∅ hyperplane H containing itself x2 x1. Now let us say that (x1, x2) int(K) int(K) ∆ is a − n ∈ × \ cylindrical pair for K if StrcvxK (x1, x2) = Grn−1(R ). Here ∆ denotes the diagonal subset. We denote by Cylpair(K) int(K) int(K)6 ∆ the subset of cylindrical pairs associated to K. The term cylindrical pair is⊂ justified by× the following\ result. n Lemma 3.4. For any (x1, x2) Cylpair(K), we can find a hyperplane H Grn−1(R ) and a convex body C of H such that ∈ ∈ H (K x) = C ∩ − for any x [x1, x2]. Equivalently, we have that (H + [x1, x2]) K = C + [x1, x2]. ∈ ∩

Proof. Picking H/ StrcvxK (x1, x2) we get that ∈ fK (H, λx1 + (1 λ)x2) = λfK (H, (x1) + (1 λ)fK (H, x2) − − n−2 for λ = 1/2 and thus for all λ (0, 1). It follows by the proof of Lemma 3.3 that u S (H) and λ [0, 1] ∈ ∀ ∈ ∀ ∈ hH∩(K−x1)(u) = hH∩(K−λx1−(1−λ)x2)(u). Therefore there exists a convex body C of H such that H (K z) = C for any z [x1, x2]. ∩ − ∈ Equivalently, we have that (H + [x1, x2]) K = C + [x1, x2]. ∩ 

The following result describes how cylindrical pairs of K should interact with the support of µ∂B to ensure strict convexity for the Holmes-Thompson area. Recall that the support supp(µ) of a Borel measure µ is the intersection of all closed sets of total measure.

Proposition 3.5. The map x AK−x(∂B) is strictly convex on int(K) if and only if supp(µ∂B) 7→ ∩ StrcvxK (x1, x2) = for every pair (x1, x2) Cylpair(K). 6 ∅ ∈ 8 FLORENT BALACHEFF, GIL SOLANES, AND KROUM TZANEV

Observe that supp(µ∂B) StrcvxK (x1, x2) = for every (x1, x2) / Cylpair(K) as StrcvxK (x1, x2) = n ∩ 6 ∅ ∈ Grn−1(R ) in this case. So we can replace “for every pair (x1, x2) Cylpair(K)” by “for every ∈ x1 = x2 int(K)” in the proposition above. 6 ∈

Proof. The integral of a non-negative continuous function against a Borel measure is positive if and only if the support of the measure meets the interior of the support of the function. Therefore, fixing x1 = x2 int(K), we get that 6 ∈

AK−x (∂B) + AK−x (∂B) 2 A (∂B) 1 2 − K−(x1+x2)/2 1 = [fK (H, x1) + fK (H, x2) 2 fK (H, (x1 + x2)/2)] dµ∂B(H) ε n − n−1 ZGrn−1(R ) is positive if and only if StrcvxK (x1, x2) meets supp(µ∂B). 

We are now ready to prove Theorem 3.1.

Proof of Theorem 3.1. By Proposition 3.5, the map x AK−x(∂B) is always strictly convex on n 7→ 1 int(K) when supp(µ∂B) = Grn−1(R ) which holds if ∂B is of class C .  2 Remark 3.6. We now present an example of two convex bodies K and B in the plane R such that the functional x AK−x(∂B) is convex but not strictly convex: 7→

( 3, 1) (3, 1) (0, 1) −

( 1, 0) (1, 0) − B ( 3, 1) (3, 1) (0, 1) − − K − −

Observe that supp(µ∂B) consists of the two directions x = y, x = y each of weight 2√2. It is easy to check that the interior of the shaded area corresponds to points− (x, y) where the functional t AK−(x+t,y)(∂B) is constant for small values of t. By symmetry we deduce that this functional achieves7→ its minimum on any point lying on the black segment.

Next, we describe an optimal property on B that ensures strict convexity for any convex body K. n For this, let us define for any L Gr1(R ) the subset ⊂ n Star(L) := H Grn−1(R ) L H . { ∈ | ⊂ } Proposition 3.7. Let B be a convex body. The map x int(K) AK−x(∂B) is strictly convex for ∈ 7→ n any convex body K if and only if supp(µ∂B) Star(L) = for any L Gr1(R ). ∩ 6 ∅ ∈

Proof. Recall that as already observed, for a fixed H and K, we know by [Santal´o1949] that for any x0 int(K) the map x fK (H, x) is strictly convex on (x0 + H) int(K). In particular, for any ∈ 7→ ∩ x1 = x2 int(K), we have that Star(span(x2 x1)) StrcvxK (x1, x2). So if supp(µ∂B) Star(L) = 6 ∈ n − ⊂ ∩ 6 ∅ for any L Gr1(R ), we directly get that supp(µ∂B) StrcvxK (x1, x2) = for any convex body K ∈ ∩ 6 ∅ and x1 = x2 int(K). By Proposition 3.5, it ensures that the map x int(K) AK−x(∂B) is always strictly6 ∈ convex. ∈ 7→ n Now suppose that supp(µ∂B) Star(L) = for some L Gr1(R ). ∩ ∅ ∈ Lemma 3.8. The family of open sets StrcvxK (x1, x2) where K runs over convex bodies and { }(K,x1,x2) x1, x2 are any pair of distinct points of int(K) such that span(x2 x1) = L is an open neighborhood base of Star(L). − HOLMES-THOMPSON AREA & SANTALO´ POINT 9

n Proof of the Lemma. Fix an open neighborhood U of Star(L) in Grn−1(R ). We construct a con- vex body K such that for some x1 = x2 int(K) with span(x2 x1) = L we have Star(L) 6 ∈ − ⊂ StrcvxK (x1, x2) U. ⊂

K

H K ∩ H U c [∈ e

ε0 K

e

0. F

H K ∩ e

ε0 H

For this consider the (possibly empty) compact set U c and let K be any infinite cylinder with direction L, with a compact convex base and that contains 0 in its interior. Now fix ε0 > 0 and a unit vector v L. As U c is compact and disjoint from Star(L), wee can consider a finite subcylinder ∈ K of K delimited by two affine hyperplanes whose underlying directions are not in Star(L) and such c that (H + εv) K = (H + εv) K for any H U and any ε ε0. By taking x1 = ε0v and ∩ ∩ c ∈ | | ≤ − x2 = εe0v we have that for any H U , the map x H (K x) is constant on [x1, x2] and so ∈ 7→ ∩ − H/ StrcvxK (x1, xe2). ∈ 

c Lemma 3.8 applied to the open neighborhood U := supp(µ∂B) implies that there exist K and c x1 = x2 int(K) such that StrcvxK (x1, x2) supp(µ∂K ) and by Proposition 3.5 we are done. 6 ∈ ⊂ 

To conclude this section, we describe an optimal property on K that ensures strict convexity for any convex body B.

Proposition 3.9. Then the map x AK−x(∂B) is strictly convex on int(K) for any convex body B if and only if K has no cylindrical7→ directions, that is Cylpair(K) = . ∅

Proof. The fact that the condition Cylpair(K) = is sufficient follows directly from 3.5. To prove that ∅ n it is also necessary, let K be a convex body with StrcvxK (x1, x2) = Grn−1(R ) for some cylindrical 6 pair (x1, x2). We will construct B such that supp(µ∂B) StrcvxK (x1, x2) = . For this we need the following lemma. ∩ ∅ c Lemma 3.10. If StrcvxK (x1, x2) is nonempty, then it has a nonempty interior.

n Proof. According to Lemma 3.4, if StrcvxK (x1, x2) = Grn−1(R ), we can find a hyperplane H0 n 6 ∈ Grn−1(R ) and a convex body C0 of H0 such that H0 (K x) = C0 for any x [x1, x2]. Fix any 0 0 0 0 ∩ − ∈ proper subinterval [x , x ] of [x1, x2], that is [x , x ] ]x1, x2[. We can find an open neighbourhood 1 2 1 2 ⊂ U of H0 such that for any H U there is a convex body C of H such that H (K x) = C 0 0 ∈ 0 0 ∩ − for any x [x1, x2]. In particular H/ StrcvxK (x1, x2) StrcvxK (x1, x2) which implies that ∈ c ∈ ⊂ StrcvxK (x1, x2) contains an open neighborhood of H0. 

n c From this lemma, it follows that if StrcvxK (x1, x2) = Grn−1(R ), then StrcvxK (x1, x2) contains n linearly independent hyperplanes. So, for any parallelotope6 B whose face directions are precisely these n hyperplanes, we have that supp(µ∂B) StrcvxK (x1, x2) = , and so we can conclude by ∩ ∅ Proposition 3.5.  10 FLORENT BALACHEFF, GIL SOLANES, AND KROUM TZANEV

3.2. Proof of the properness. It remains to show that the map is proper on the interior of K. This holds even when the convex body B is no longer of class C1, and easily follows from the well-known fact that the map x K−x(B) is proper on the interior of K and the following generalization of the classical isoperimetric7→ V inequality to asymmetric Minkowski spaces.

Proposition 3.11 (Isoperimetric inequality for Holmes-Thompson volume). Let B and K be two convex bodies of a n-dimensional real vector space V . Suppose that K contains the origin as an interior point. Then n n AK (∂B) (4n) n−1 . K (B) ≥ n!εn V This inequality is stated in [Thompson 1996, Theorem 6.6.4] in the case where K is symmetric. The proof straightforward generalizes to the non-symmetric case so we briefly survey the arguments.

n ∗ Proof. After fixing some isomorphism V R , we identify the dual body A of any convex body A containing the origin as an interior point' with its polar body A◦. First recall that (compare with [Thompson 1996, Theroem 5.2.2])

AK (∂B) = n V (B[n 1],IK ) · − where V denotes the mixed volume and IK the isoperimetrix convex body defined as the unique symmetric convex body with support function

◦ ⊥ ◦ hI (u) = π ⊥ (K ) /εn−1 = (K u ) /εn−1. (3.3) K | u | | ∩ | Observe that IK is a zonoid according to [Schneider 2013, Formula 5.80 and Theorem 3.5.3]. Next we check that n n AK (∂B) AK (∂IK ) n−1 n−1 . K (B) ≥ K (IK ) V V ◦ Indeed as AK (∂IK ) = n IK and K (B) = B K /εn, the above inequality is equivalent to | | V | || | n n−1 V (B[n 1],IK ) IK B − ≥ | || | which directly follows from Minkowski’s inequality, see [Schneider 2013, Theorem 7.2.1].

Now recall that the normalized isoperimetrix convex body I˜K is defined as the unique dilated of the isoperimetrix satisfying n K (I˜K ) = AK (∂I˜K ). Therefore V n ˜ n ◦ ˜ AK (∂B) AK (∂IK ) n n K IK = n K (I˜K ) = n | || |. n−1 n−1 K (B) ≥ K (I˜K ) V εn V V ◦ Besides it is easy to see that in fact I˜K = εn IK / K . Using equation (3.3), we also see that · | | ◦ εn−1 hIK is precisely the support function of the projection body ΠK . Consequently we find the · ◦ ◦ identity I˜K = (εn/εn−1 K ) ΠK . Therefore | | · ◦ n n ◦ εn−1 K ◦ ◦ εn−1 ◦ ◦ n−1 ◦ ◦ (I˜K ) = | | (ΠK ) = K K (ΠK ) , | | ε | | ε | || | | |  n   n  which implies together with Petty’s projection inequality [Petty 1971], see [Schneider 2013, Formula 10.86], that ◦ ◦ (I˜K ) K . | | ≤ | | Finally n ˜ ◦ ˜ n AK (∂B) n (IK ) IK (4n) n−1 n | || | K (B) ≥ εn ≥ n!εn V using Reisner optimal lower bound on the volume product for zonoids [Reisner 1985].  HOLMES-THOMPSON AREA & SANTALO´ POINT 11

4. Equiaffine differential geometry

We now introduce classical notions from equiaffine geometry and some notations that will be used to prove Theorem 1.3 in the next two sections. From here on, let us suppose the n-dimensional vector space V endowed with a non-trivial alternate multilinear n-form, which we denote by det. Let M V be a hypersurface and take a vector field Ξ transverse to M. Then, for every pair of tangent vector⊂ fields X,Y on M, we can decompose the flat connection of V as ∇ M X Y = Y + g(X,Y ) Ξ (4.1) ∇ ∇X · where M is an affine torsion-free connection on M and g is a field of symmetric bilinear forms. The hypersurface∇ M is said to be non-degenerate if g is nowhere degenerate, a condition that does not depend on Ξ. Definition 4.1. For each non-degenerate oriented hypersurface M, there is a unique transversal vector field Ξ, called equiaffine normal field (or Blaschke’s normal field), such that (cf. [Nomizu & Sasaki 1994, Ch.II, Thm.3.1])

i) vΞ TxM for every x M and v TxM, ii) ∇the volume∈ (n 1)-form∈α associated∈ to g satisfies − α(ξ1, . . . , ξn−1) = det(Ξ(x), ξ1, . . . , ξn−1)

for every x M and ξ1, . . . , ξn−1 TxM. ∈ ∈ The pseudo-riemannian metric g and its volume form α are called equiaffine metric and equiaffine area measure respectively. We assume from here on that g is Riemannian.

We will denote by E = E(M) the frame bundle of M whose fiber over x M is ∈ n−1 Ex = (x, ξ1, . . . , ξn−1) M V : span(ξ1, . . . , ξn−1) = TxM . { ∈ × } Given (x, ξ1, . . . , ξn−1) E, items i) and ii) in Definition 4.1 read ∈ 2 det( Ξ, ξ1, . . . , ξn−1) = 0, det(Ξ(x), ξ1, . . . , ξn−1) = det(g(ξi, ξj))i,j. (4.2) ∇ Definition 4.2. Given an oriented hypersurface M V and ξ E let for any i, j = 1, . . . , n 1 ⊂ ∈ − Li,j(ξ) := det( ξ Xj, ξ1, . . . , ξn−1) ∇ i where Xj is any tangent vector field with Xj(x) = ξj. It is easy to check that Li,j = Lj,i and does ∞ not depend on Xj. Let L C (E) be the function defined by ∈ L(ξ) := det(Li,j(ξ))i,j.

Proposition 4.3. For any ξ = (x, ξ1, . . . , ξn−1) E we have ∈ n+1 L(ξ) = det(Ξ(x), ξ1, . . . , ξn−1) . (4.3)

Proof. We can assume ξ1, . . . , ξn−1 to be positively oriented, as both sides of (4.3) are equally { } affected by a permutation. Indeed, the effect on the matrix (Li,j)i,j of a transposition of ξ1, , ξn−1 ··· is a sign change of all entries Li,j, and a simultaneous permutation of two lines and two columns. From (4.1) and (4.2) we get

Lk,l(ξ) = det( ξ Xl, ξ1, . . . , ξn−1) ∇ k = g(ξk, ξl) det(Ξ(x), ξ1, . . . , ξn−1) (4.4) 1 2 = g(ξk, ξl) det(g(ξi, ξj))i,j. (4.5) It follows using (4.2) again that n−1 1+ 2 n+1 L(ξ) = det(Lk,l(ξ)) = det(g(ξk, ξl))k,l = det(Ξ(x), ξ1, . . . , ξn−1) . (4.6)  12 FLORENT BALACHEFF, GIL SOLANES, AND KROUM TZANEV

Remark 4.4. By (4.5) and (4.6), the equiaffine metric is simply given by − 1 g(ξi, ξj) = L(ξ) n+1 Li,j(ξ). | | 1 In turn, the equiaffine normal vector can be obtained from g as Ξ = n−1 ∆f where ∆ is the Laplacian with respect to g and f : M V is the inclusion (cf. [Nomizu & Sasaki 1994, Thm.6.5, Ch. II]). We will not make use of this fact.→

5. Measure of lines in terms of hyperplanes

In this section we use the boundary sphere ∂B to construct a parametrization space for the oriented affine lines of the dual vector space in terms of intersections of affine hyperplanes. We then show how to rewrite the measure involved in Alvarez´ Paiva formula in this new space. Recall that we have fixed some non-trivial alternate multilinear n-form det on the n-dimensional vector space V . Let B be a smooth quadratically convex body of V that contains the origin as an interior point. In particular its dual body is smooth, and the Legendre transform : ∂B ∂B∗, uniquely defined by L → ker (x) = Tx∂B and (x), x = 1, L hL i is a well defined diffeomorphism. Moreover, ∂B is non-degenerate in the sense of the previous section. We orient ∂B with the inward vector (i.e. as the boundary of V B) so that the equiaffine metric g \ is positive definite. Associated to the hypersurface ∂B we also have well defined functions Li,j and L C∞(E(∂B)) (see Definition 4.2). ∈ Consider a smooth local section ζ : U ∂B E(∂B) ⊂ → x ζ(x) = (x, ζ1(x), . . . , ζn−1(x)). 7→ We define the following diagram

n−1 F (R 0 ) U E(∂B) \{ } × G H ∗ π ∗ V ∂B G+(V ) × where

∗ ∗ G+(V ) is the space of oriented affine lines in V introduced in section 2.3; • n−1 for λ = (λ1, . . . , λn−1) (R 0 ) and x U • ∈ \{ } ∈ F (λ, x) = (x, λ1ζ1(x), . . . , λn−1ζn−1(x));

G(x, ξ1, . . . , ξn−1) is the affine line • ∗ p V : ξ1, p = = ξn−1, p = 1 { ∈ h i ··· h i } oriented by (x); L 1 n−1 ∗ for (x, ξ1, , ξn−1) E(∂B) let (x), ξ , . . . , ξ be the dual basis in V of the basis • ··· ∈ {L } x, ξ1, . . . , ξn−1 , and set { } n−1 i p(x, ξ1, . . . , ξn−1) = ξ . i=1 X Then define H(x, ξ1, . . . , ξn−1) = (p(x, ξ1, . . . , ξn−1), x); π is the projection map • π(p, x) = p + (x) +. hL i HOLMES-THOMPSON AREA & SANTALO´ POINT 13

It is easy to check that the previous diagram commutes.

Indeed observe that ξi, p(x, ξ1, . . . , ξn−1) = 1 and ξi, (x) = 0 for all i, which leads to the com- mutativity property:h i h L i

π H(x, ξ1, . . . , ξn−1) = p(x, ξ1, . . . , ξn−1) + (x) + = G(x, ξ1, . . . , ξn−1). ◦ hL i ∗ Recall that G+(V ) is endowed with a symplectic form ωB∗ (see section 2.3). It will be convenient ∗ to consider on G+(V ) the following associated volume element n(n−1) n−1 ηB∗ = ( 1) 2 ω ∗ − B and take the corresponding orientation. Our goal in this section is to compute the pull-back form ∗ G ηB∗ . For this, we introduce the following 2-forms.

Definition 5.1. Let ω1, . . . , ωn−1 be the 2-forms on E(∂B) given by

ωi = det(dπi, dπi, ξ1 ξi,..., ξ\i ξi, . . . , ξn−1 ξi), (x,ξ1,...,ξn−1) − − − − where

πi : E(∂B) V → (x, ξ1, . . . , ξn−1) ξi. 7→ ∗ Using these 2-forms we are able to express G ηB∗ as follows. Proposition 5.2. ∗ (n 1)! G ηB∗ = − ω1 ωn−1. (5.1) L ∧ · · · ∧

The rest of this section is devoted to the proof of the proposition above.

5.1. Technical lemmas. Let us begin with the following computation. Lemma 5.3.

∗ ∗ (n 1)! 1 n−1 (F G η ∗ ) = dλ dλ ζ (x) ζ (x) B (λ,x) n−1 2 1 n−1 i=1 λi ∧ · · · ∧ ∧ ∧ · · · ∧ 1 n−1 where (x), ζ (x), . . . , ζ (x) is the dual basis of x, ζ1(x), . . . , ζn−1(x) . {L } Q { } Proof. Note first that ∗ ∗ ∗ ∗ ∗ ∗ ∗ F G ωB∗ = F H π ωB∗ = F H ωV ∗ ∗ ∗ where we have abusively denoted i ωV ∗ by ωV ∗ (here i stands for the canonical inclusion V ∂B , V ∗ V ). × → × Fixing x0 U we write ∈ n−1 i ωV ∗ = x0 (x0) + ζi(x0) ζ (x0) ∧ ∧ L i=1 X globally on V ∗ V . × n−1 i Since H F (λ, x) = (p F (λ, x), x) = ( ζ (x)/λi, x) we have ◦ ◦ i=1 i 2 d(H F ) (∂/∂λi) = ζ (x)/λ , 0 and d(H F ) (ζi(x)) = ( , ζi(x)). ◦ (λ,x) − P i ◦ (λ,x) ∗ i j Thus, modulo terms of the formζ (x0) ζ (x0), ∧ n−1 ∗ ∗ i (F H ωV ∗ ) ωV ∗ (d(H F ) (∂/∂λi) , d(H F ) (ζi(x0))) dλi ζ (x0) (λ,x0) ≡ ◦ (λ,x0) ◦ (λ,x0) ∧ i=1 X n−1 1 i = dλi ζ (x0). − λ2 ∧ i=1 i X The statement follows.  14 FLORENT BALACHEFF, GIL SOLANES, AND KROUM TZANEV

Secondly we prove the following identity. Lemma 5.4. n−1 ∗ n−1 1 n−1 F (ω1 ωn−1) = L(ζ(x)) λ dλ1 dλn−1 ζ (x) ζ (x). ∧ · · · ∧ (λ,x) i ∧ · · · ∧ ∧ ∧ · · · ∧ i=1 ! Y

n−1 Proof. Fix x0 U and put ξi = ζi(x0). Considering E(∂B) V V we have ∈ ⊂ ×

dF(λ,x0) (∂/∂λj) = (0 ; 0,..., 0, ξj, 0,..., 0), and

dF(λ,x0)(ξj) = (ξj ; λ1(dζ1)x0 (ξj), . . . , λn−1(dζn−1)x0 (ξj)). Thus ∗ ∂ ∂ F ωi , = 0 ∂λ ∂λ  j k  for any i, j, k and

∗ ∂ F ωi , ξ = 0 ∂λ k  j  for any i, j, k such that i = j, while 6 ∗ ∂ F ωi , ξj = det(ξi, λi(dζi)x (ξj), λ1ξ1 λiξi,..., λiξ\i λiξi, . . . , λn−1ξn−1 λiξi) ∂λ − 0 − − −  i  = det(ξi, λi(dζi)x (ξj), λ1ξ1,..., λiξi, . . . , λn−1ξn−1) − 0 n−1 i+1 = ( 1) λk det( ξ ζi, ξ1d, . . . , ξn−1) − ∇ j k=1 ! Y that is n−1 ∗ ∂ i+1 F ωi , ξj = ( 1) λk Li,j(x0, ξ1, . . . , ξn−1). ∂λi − k=1 !   Y j j Therefore, putting Li,j = Li,j(x0, ξ1, . . . , ξn−1), and ξ = ζ (x0), we have ∗ F (ω1 ωn−1) ∧ · · · ∧ (λ,x0) n−1 n−1 n−1 n−1 (n−1)n n−1 j j = ( 1) ( 1) 2 λ L1,jdλ1 ξ Ln−1,jdλn−1 ξ − − i  ∧  ∧ · · · ∧  ∧  i=1 ! j=1 j=1 Y X X n−1     n−1 j1 jn−1 = λ dλ1 dλn−1 L1,j Ln−1,j ξ ξ i ∧ · · · ∧ ∧  1 ··· n−1 ∧ · · · ∧  i=1 ! j ,...,j Y 1 Xn−1 n−1   n−1 1 n−1 = λ det(Li,j)i,j dλ1 dλn−1 ξ ξ k ∧ · · · ∧ ∧ ∧ · · · ∧ k=1 ! Y from which we get the announced formula. 

5.2. Proof of Proposition 5.2. Let us denote (n 1)! Ω := − ω1 ωn−1. L ∧ · · · ∧ The proposition will follow easily after proving that

∗ ∗ ∗ (1) F Ω = F G ηB∗ ; (2) iX Ω = 0 for all X ker dG. ∈ HOLMES-THOMPSON AREA & SANTALO´ POINT 15

Indeed, given ξ = (x, ξ1, . . . , ξn−1) E(∂B) we can take a local section ζ of E(∂B) such that ζ(x) = ξ. In particular the map F associated∈ to this section satisfies that F ((1,..., 1), x) = ξ. Then we observe that im dF ker dGξ = TξE(∂B), ((1,...,1),x) ⊕ because G F is a diffeomorphism according to Lemma 5.3. Equality (5.1) follows directly using points (1) and◦ (2). Let us check point (1). Note that (e.g. by Proposition 4.3) n−1 n+1 L F (λ, x) = λ L(x, ζ1(x), . . . , ζn−1(x)). (5.2) ◦ i i=1 ! Y Hence, by Lemma 5.4, at ξ = F (λ, x) we have

∗ (n 1)! ∗ F Ω = − F (ω1 ωn−1) L F (λ, x) ∧ · · · ∧ ◦ (n 1)! = dλ dλ ζ1(x) ζn−1(x), n−1 2 1 n−1 i=1 λi ∧ · · · ∧ ∧ ∧ · · · ∧ ∗ ∗ which is precisely F G ηB∗ byQ Lemma 5.3. In order to check point (2), we easily verify that n−1 ker dGξ = (0,Zij) V (V ) i, j 1, . . . , n 1 with i = j (5.3) h ∈ × | ∈ { − } 6 i where Zij = (0,..., 0, ξj ξi , 0,..., 0) V. − ∈ ith position Indeed we have G(x, ξ1, . . . , ξi−1, (1 t)ξi + tξ|j,{z ξi+1}, . . . , ξn−1) = G(x, ξ1, . . . , ξn−1) for all t and differentiating the curve − t (ξ1,..., (1 t)ξi + tξj, . . . , ξn−1) 7→ − at time 0 gives the vector Zij. The linear independence of (ξ1, . . . , ξn−1) then ensures that the family Zij i, j 1, . . . , n 1 with i = j is also linearly independent. Its cardinality is (n 1)(n 2), which{ | coincides∈ { with− dim} ker dG =6 dim} E(∂B) 2(n 1) as G F is a diffeomorphism.− This shows− (5.3). − − ◦ Next we see that if i = k, then 6 (i ωk)ξ = det(dπk(Zij), dπk, ξ1 ξk,..., ξ\k ξk, . . . , ξn−1 ξk) = 0 (0,Zij ) − − − − and that

(i ωi)ξ = det(dπi(Zij), dπi, ξ1 ξi,..., ξ\i ξi, . . . , ξn−1 ξi) (0,Zij ) − − − − = det(ξj ξi, dπi, ξ1 ξi,..., ξ\i ξi, . . . , ξn−1 ξi) − − − − − = 0, which proves point (2). 

6. First variation of the dual area under translations

Recall that we have chosen some non-trivial alternate multilinear n-form denoted by det on the n- dimensional vector space V and that B is a smooth quadratically convex body of V that contains the origin as an interior point. In particular the normed vector space (V, B) is Minkowski. Let K denote another convex body of V . k · k In this section we prove Theorem 1.3 by explicitly computing

d ∗ AB∗ (∂(K tv) ) dt − t=0 for any v V . ∈ 16 FLORENT BALACHEFF, GIL SOLANES, AND KROUM TZANEV

6.1. Preliminaries. Given v V , let T v : E(∂B) E(∂B) be given by ∈ → T v(x, ξ1, . . . , ξn−1) = (xv, ξ1 + v, . . . , ξn−1 + v).

Here xv ∂B is uniquely determined by the conditions that ξi + v Tx ∂B for all i and that the ∈ ∈ v basis xv, ξ1 + v, . . . , ξn−1 + v is negatively oriented. { } ∗ ∗ Let Tv : G+(V ) G+(V ) be defined by → Tv(G(ξ)) = G(T v(ξ)). 0 0 To check that this is independent of ξ, note that G(ξ) = G(ξ ) implies that each ξi is an affine combination of ξ1, . . . , ξn−1. Indeed recall that G(ξ) is the affine line ∗ p V : ξ1, p = = ξn−1, p = 1 { ∈ h i ··· h i } 0 0 oriented by (x). Clearly, the same holds for ξ +v and the ξj +v which yields G(T v(ξ)) = G(T v(ξ )) L i up to the orientation. To check orientations we note that v xv is continuous and hence the 0 7→ orientations of G(T v(ξ)) and G(T v(ξ )) respectively agree or disagree for all v; and they coincide for v = 0. ∗ Lemma 6.1. For v V and L G+(V ) we have ∈ ∈ ∗ ∗ L (K v) = Tv(L) K = . ∩ − 6 ∅ ⇐⇒ ∩ 6 ∅

Proof. Let L = G(ξ) = G(x, ξ1, . . . , ξn−1) and note that ∗ ∗ ∗ G(ξ) (K v) = p V : ξi, p = 1 (K v) = , i ∩ − 6 ∅ ⇐⇒ { ∈ h i } ∩ − 6 ∅ ∀ ξi / K v, i. ⇐⇒ ∈ − ∀ ∗ By the same token, G(T v(ξ)) intersects K if and only if ξi + v / K for all i, which is obviously ∈ equivalent to the condition just obtained above. 

Then by Proposition 2.1 and Lemma 6.1,

∗ 1 AB∗ (∂(K tv) ) = ηB∗ − (n 1)!εn−1 ∗ ∗ − Z{T−tv(L): L∈G+(V ),L∩K 6=∅} 1 ∗ = T−tv ηB∗ . (n 1)!εn−1 ∗ ∗ − Z{L∈G+(V ): L∩K 6=∅}

Consider the vector field Zv on E(∂B) given by d Zv(x, ξ1, . . . , ξn−1) = T tv(x, ξ1, . . . , ξn−1), dt t=0 and recall that

∗ (n 1)! G ηB∗ = Ω = − ω1 ωn−1. L ∧ · · · ∧ n−1 Let now Ui be a finite cover of ∂B, take a local section ζi of each E(Ui), and let Fi :(R 0 ) { } \{ } × Ui E(∂B) be the corresponding map. Since G Fi is a diffeomorphism onto its image, taking a → ◦ partition of unity ρi subordinate to Ui , we have { } ∗ 1 ∗ ∗ ∗ AB∗ (∂(K tv) ) = ρi Fi G T−tv ηB∗ − (n 1)!ε −1 ∗ · n−1 i (G◦Fi) ({L: L∩K 6=∅}) − X Z 1 ∗ ∗ = ρi Fi T −tv Ω (n 1)!ε −1 ∗ · n−1 i (G◦Fi) ({L: L∩K 6=∅}) − X Z and we deduce that

d ∗ 1 ∗ ∗ AB (∂(K tv) ) = ρi Fi Zv Ω. (6.1) dt − −(n 1)!εn−1 −1 ∗ · L t=0 i Z(G◦Fi) ({L: L∩K 6=∅}) − X

HOLMES-THOMPSON AREA & SANTALO´ POINT 17

Proposition 6.2. For any ξ = (x, ξ1, . . . , ξn−1) E(∂B) and any v V ∈ ∈

Zv L Z Ω = (n 1)! L ω1 ωn−1 L v − − L2 ∧ · · · ∧ and n−1 n Zv L(ξ) = (n + 1) (det(Ξ(x), ξ1, . . . , ξn−1)) det(Ξ(x), ξ1, . . . , ξi−1, v, ξi+1, . . . , ξn−1) L i=1 ! X where Ξ is the interior equiaffine normal field of ∂B.

Proof. The first equality follows from Z ωi = 0. As for the second, by Proposition 4.3 we have L v n+1 L(T v(ξ)) = L(xtv, ξ1 + tv, . . . , ξn−1 + tv) = det(Ξ(xtv), ξ1 + tv, . . . , ξn−1 + tv) . Hence

n d Z L(ξ) = (n + 1) det(Ξ(x), ξ1, . . . , ξn−1) det(Ξ(xtv), ξ1 + tv, . . . , ξn−1 + tv). L v dt t=0 d d The proposition follows by noting that Ξ(xtv) = X Ξ(x) with X = xtv which according dt t=0 ∇ dt t=0 to the first identity of (4.2) implies that

n−1 d det(Ξ(xtv), ξ1 + tv, . . . , ξn−1 + tv) = det (Ξ(x), ξ1, . . . , ξi−1, v, ξi+1, . . . , ξn−1) . dt t=0 i=1 X



6.2. Proof of Theorem 1.3. Take a local section ζ : U E(∂B) mapping each x U to a → ∈ positive orthonormal basis e1(x), . . . , en−1(x) of Tx∂B with respect to the Riemannian metric g. 1 { n−1 ∗ } For all x U let e (x), . . . , e (x), Ξ (x) denote the dual basis of e1(x), . . . , en−1(x), Ξ(x) . ∈∗ { } ∗ { } Since ker Ξ (x) = span(e1(x), . . . , en−1(x)), we have Ξ (x) (x) . Take the smooth measure 1 n−1 ∈ hL i ∗ µ = e e on U. For each x U the dual measure νx on ker Ξ(x) T ∂B of the Lebesgue ∧ · · · ∧ ∈ ' x measure µx is given by νx = e1(x) en−1(x). ∧ · · · ∧ First observe that if F denotes the map corresponding to ζ we have

L F (λ, x) = L(x, λ1e1(x), . . . , λn−1en−1(x)) ◦ n−1 n+1 n+1 = λi det(Ξ(x), e1(x), . . . , en−1(x)) i=1 ! Y n−1 n+1 = λi i=1 ! Y using the fact that e1(x), . . . , en−1(x) is a positive orthonormal basis together with point ii) in Definition 4.1, and that{ similarly } n−1 n−1 n+1 1 i Z L F (λ, x) = (n + 1) λ e (x), v v ◦ i λ h i L i=1 ! i=1 i Y X where , denotes the canonical duality pairing. Furthermore by Lemma 5.4 the following holds: h· ·i n−1 ∗ n−1 1 n−1 F (ω1 ωn−1) = λ dλ1 dλn−1 e (x) e (x). ∧ · · · ∧ (λ,x) i ∧ · · · ∧ ∧ ∧ · · · ∧ i=1 ! Y Therefore, using Proposition 6.2, we get the following formula: n−1 n−1 ∗ (n + 1)! 1 i 1 F ( Z Ω) = e (x), v dλ1 dλn−1 dµ(x). v (λ,x) − n λ λ2 ∧ · · · ∧ ∧ L * i=1 i + i=1 i ! X Y 18 FLORENT BALACHEFF, GIL SOLANES, AND KROUM TZANEV

Let us define the map n−1 ∗ px :(R 0 ) ker Ξ(x) V \{ } → ⊂ n−1 1 i (λ1, . . . , λn−1) e (x) 7→ λ i=1 i X which is a diffeomorphism onto its image that satisfies

G F (λ, x) = px(λ) + (x) +. ◦ hL i In particular, we have −1 ∗ n−1 ∗ (G F ) ( L: L K = ) = (λ, x) (R 0 ) U : G F (λ, x) K = ◦ { ∩ 6 ∅} { ∈ \{ } × ◦ ∩ 6 ∅} n−1 ∗ = (λ, x) (R 0 ) U : px(λ) πx(K ) { ∈ \{ } × ∈ } ∗ where πx : V ker Ξ(x) is the linear projection with ker πx = (x) . → hL i A simple computation shows that n−1 −1 1 (p )∗ νx = dλ1 dλn−1 x λ2 ∧ · · · ∧ i=1 i ! Y as measures. Therefore ∗ F Zv Ω −1 ∗ L Z(G◦F ) ({L: L∩K 6=∅}) (n + 1)! −1 = px(λ), v d(px )∗(νx) dµ(x) − n −1 ∗ h i Zx∈U Zλ∈px (πx(K )) ! (n + 1)! = q, v dνx(q) dµ(x) − n ∗ h i Zx∈U Zq∈πx(K ) ! (n + 1)! = q dνx(q) dµ(x), v . − n ∗ *Zx∈U Zπx(K ) ! + Covering ∂B with local sections and using (6.1), we deduce the stated formula:

d ∗ n + 1 AB∗ (∂(K tv) ) = q dνx(q) dµ(x), v .  dt − ε ∗ t=0 n−1 *Zx∈U Zπx(K ) ! +

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Florent Balacheff, Universitat Autonoma` de Barcelona, Spain. Email address: [email protected]

Gil Solanes, Universitat Autonoma` de Barcelona and Centre de Recerca Matematica,` Spain. Email address: [email protected]

Kroum Tzanev, Universite´ de Lille, France. Email address: [email protected]