An Invitation to Generalized Minkowski Geometry
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Fuchsian Convex Bodies: Basics of Brunn–Minkowski Theory François Fillastre
Fuchsian convex bodies: basics of Brunn–Minkowski theory François Fillastre To cite this version: François Fillastre. Fuchsian convex bodies: basics of Brunn–Minkowski theory. 2011. hal-00654678v1 HAL Id: hal-00654678 https://hal.archives-ouvertes.fr/hal-00654678v1 Preprint submitted on 22 Dec 2011 (v1), last revised 30 May 2012 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Fuchsian convex bodies: basics of Brunn–Minkowski theory François Fillastre University of Cergy-Pontoise UMR CNRS 8088 Departement of Mathematics F-95000 Cergy-Pontoise FRANCE francois.fi[email protected] December 22, 2011 Abstract The hyperbolic space Hd can be defined as a pseudo-sphere in the ♣d 1q Minkowski space-time. In this paper, a Fuchsian group Γ is a group of linear isometries of the Minkowski space such that Hd④Γ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be defined, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn–Minkowski theory. -
Combination of Cubic and Quartic Plane Curve
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 43-53 www.iosrjournals.org Combination of Cubic and Quartic Plane Curve C.Dayanithi Research Scholar, Cmj University, Megalaya Abstract The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid. A cross-section of the set of unistochastic matrices of order three forms a deltoid. The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid. The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six. The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856. The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides. I. Introduction Various combinations of coefficients in the above equation give rise to various important families of curves as listed below. 1. Bicorn curve 2. Klein quartic 3. Bullet-nose curve 4. Lemniscate of Bernoulli 5. Cartesian oval 6. Lemniscate of Gerono 7. Cassini oval 8. Lüroth quartic 9. Deltoid curve 10. Spiric section 11. Hippopede 12. Toric section 13. Kampyle of Eudoxus 14. Trott curve II. Bicorn curve In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis. -
Using Elimination to Describe Maxwell Curves Lucas P
Louisiana State University LSU Digital Commons LSU Master's Theses Graduate School 2006 Using elimination to describe Maxwell curves Lucas P. Beverlin Louisiana State University and Agricultural and Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_theses Part of the Applied Mathematics Commons Recommended Citation Beverlin, Lucas P., "Using elimination to describe Maxwell curves" (2006). LSU Master's Theses. 2879. https://digitalcommons.lsu.edu/gradschool_theses/2879 This Thesis is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Master's Theses by an authorized graduate school editor of LSU Digital Commons. For more information, please contact [email protected]. USING ELIMINATION TO DESCRIBE MAXWELL CURVES A Thesis Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial ful¯llment of the requirements for the degree of Master of Science in The Department of Mathematics by Lucas Paul Beverlin B.S., Rose-Hulman Institute of Technology, 2002 August 2006 Acknowledgments This dissertation would not be possible without several contributions. I would like to thank my thesis advisor Dr. James Madden for his many suggestions and his patience. I would also like to thank my other committee members Dr. Robert Perlis and Dr. Stephen Shipman for their help. I would like to thank my committee in the Experimental Statistics department for their understanding while I completed this project. And ¯nally I would like to thank my family for giving me the chance to achieve a higher education. -
Minkowski Endomorphisms Arxiv:1610.08649V2 [Math.MG] 22
Minkowski Endomorphisms Felix Dorrek Abstract. Several open problems concerning Minkowski endomorphisms and Minkowski valuations are solved. More precisely, it is proved that all Minkowski endomorphisms are uniformly continuous but that there exist Minkowski endomorphisms that are not weakly- monotone. This answers questions posed repeatedly by Kiderlen [20], Schneider [32] and Schuster [34]. Furthermore, a recent representation result for Minkowski valuations by Schuster and Wannerer is improved under additional homogeneity assumptions. Also a question related to the structure of Minkowski endomorphisms by the same authors is an- swered. Finally, it is shown that there exists no McMullen decomposition in the class of continuous, even, SO(n)-equivariant and translation invariant Minkowski valuations ex- tending a result by Parapatits and Wannerer [27]. 1 Introduction Let Kn denote the space of convex bodies (nonempty, compact, convex sets) in Rn en- dowed with the Hausdorff metric and Minkowski addition. Naturally, the investigation of structure preserving endomorphisms of Kn has attracted considerable attention (see e.g. [7, 20, 29–31, 34]). In particular, in 1974, Schneider initiated a systematic study of continuous Minkowski-additive endomorphisms commuting with Euclidean motions. This class of endomorphisms, called Minkowski endomorphisms, turned out to be particularly interesting. Definition. A Minkowski endomorphism is a continuous, SO(n)-equivariant and trans- lation invariant map Φ: Kn !Kn satisfying Φ(K + L) = Φ(K) + Φ(L); K; L 2 Kn: Note that, in contrast to the original definition, we consider translation invariant arXiv:1610.08649v2 [math.MG] 22 Feb 2017 instead of translation equivariant maps. However, it was pointed out by Kiderlen that these definitions lead to the same class of maps up to addition of the Steiner point map (see [20] for details). -
Ioan-Ioviţ Popescu Optics. I. Geometrical Optics
Edited by Translated into English by Lidia Vianu Bogdan Voiculescu Wednesday 31 May 2017 Online Publication Press Release Ioan-Ioviţ Popescu Optics. I. Geometrical Optics Translated into English by Bogdan Voiculescu ISBN 978-606-760-105-3 Edited by Lidia Vianu Ioan-Ioviţ Popescu’s Optica geometrică de Ioan-Ioviţ Geometrical Optics is essentially Popescu se ocupă de buna a book about light rays: about aproximaţie a luminii sub forma de what can be seen in our raze (traiectorii). Numele autorului universe. The name of the le este cunoscut fizicienilor din author is well-known to lumea întreagă. Ioviţ este acela care physicists all over the world. He a prezis existenţa Etheronului încă is the scientist who predicted din anul 1982. Editura noastră a the Etheron in the year 1982. publicat nu demult cartea acelei We have already published the descoperiri absolut unice: Ether and book of that unique discovery: Etherons. A Possible Reappraisal of the Ether and Etherons. A Possible Concept of Ether, Reappraisal of the Concept of http://editura.mttlc.ro/iovit Ether, z-etherons.html . http://editura.mttlc.ro/i Studenţii facultăţii de fizică ovitz-etherons.html . din Bucureşti ştiau în anul 1988— What students have to aşa cum ştiam şi eu de la autorul know about the book we are însuşi—că această carte a fost publishing now is that it was îndelung scrisă de mână de către handwritten by its author over autor cu scopul de a fi publicată în and over again, dozens of facsimil: ea a fost rescrisă de la capăt times—formulas, drawings and la fiecare nouă corectură, şi au fost everything. -
Sharp Isoperimetric Inequalities for Affine Quermassintegrals
Sharp Isoperimetric Inequalities for Affine Quermassintegrals Emanuel Milman∗,† Amir Yehudayoff∗,‡ Abstract The affine quermassintegrals associated to a convex body in Rn are affine-invariant analogues of the classical intrinsic volumes from the Brunn–Minkowski theory, and thus constitute a central pillar of Affine Convex Geometry. They were introduced in the 1980’s by E. Lutwak, who conjectured that among all convex bodies of a given volume, the k-th affine quermassintegral is minimized precisely on the family of ellipsoids. The known cases k = 1 and k = n 1 correspond to the classical Blaschke–Santal´oand Petty projection inequalities, respectively.− In this work we confirm Lutwak’s conjec- ture, including characterization of the equality cases, for all values of k =1,...,n 1, in a single unified framework. In fact, it turns out that ellipsoids are the only local− minimizers with respect to the Hausdorff topology. For the proof, we introduce a number of new ingredients, including a novel con- struction of the Projection Rolodex of a convex body. In particular, from this new view point, Petty’s inequality is interpreted as an integrated form of a generalized Blaschke– Santal´oinequality for a new family of polar bodies encoded by the Projection Rolodex. We extend these results to more general Lp-moment quermassintegrals, and interpret the case p = 0 as a sharp averaged Loomis–Whitney isoperimetric inequality. arXiv:2005.04769v3 [math.MG] 11 Dec 2020 ∗Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel. †Email: [email protected]. ‡Email: amir.yehudayoff@gmail.com. 2010 Mathematics Subject Classification: 52A40. -
Equivariant Endomorphisms of the Space of Convex
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 194, 1974 EQUIVARIANTENDOMORPHISMS OF THE SPACE OF CONVEX BODIES BY ROLF SCHNEIDER ABSTRACT. We consider maps of the set of convex bodies in ¿-dimensional Euclidean space into itself which are linear with respect to Minkowski addition, continuous with respect to Hausdorff metric, and which commute with rigid motions. Examples constructed by means of different methods show that there are various nontrivial maps of this type. The main object of the paper is to find some reasonable additional assumptions which suffice to single out certain special maps, namely suitable combinations of dilatations and reflections, and of rotations if d = 2. For instance, we determine all maps which, besides having the properties mentioned above, commute with affine maps, or are surjective, or preserve the volume. The method of proof consists in an application of spherical harmonics, together with some convexity arguments. 1. Introduction. Results. The set Srf of all convex bodies (nonempty, compact, convex point sets) of ¿f-dimensional Euclidean vector space Ed(d > 2) is usually equipped with two geometrically natural structures: with Minkowski addition, defined by Kx + K2 = {x, + x2 | x, £ 7v,,x2 £ K2), Kx, K2 E Sf, and with the topology which is induced by the Hausdorff metric p, where p(7C,,7C2)= inf {A > 0 \ Kx C K2 + XB,K2 C Kx + XB], KxK2 E ®d; here B — (x E Ed | ||x|| < 1} denotes the unit ball. We wish to study the transformations of S$d which are compatible with these structures, i.e., the continuous maps $: ®d -» $$*which satisfy ®(KX+ K2) = OTC,+ <¡?K2for Kx, K2 E ñd. -
The Isoperimetrix in the Dual Brunn-Minkowski Theory 3
THE ISOPERIMETRIX IN THE DUAL BRUNN-MINKOWSKI THEORY ANDREAS BERNIG Abstract. We introduce the dual isoperimetrix which solves the isoperimetric problem in the dual Brunn-Minkowski theory. We then show how the dual isoperimetrix is related to the isoperimetrix from the Brunn-Minkowski theory. 1. Introduction and statement of main results A definition of volume is a way of measuring volumes on finite-dimensional normed spaces, or, more generally, on Finsler manifolds. Roughly speaking, a definition of volume µ associates to each finite-dimensional normed space (V, ) a norm on ΛnV , where n = dim V . We refer to [3, 28] and Section 2 for morek·k information. The best known examples of definitions of volume are the Busemann volume µb, which equals the Hausdorff measure, the Holmes-Thompson volume µht, which is related to symplectic geometry and Gromov’s mass* µm∗, which thanks to its convexity properties is often used in geometric measure theory. To each definition of volume µ may be associated a dual definition of volume µ∗. For instance, the dual of Busemann’s volume is Holmes-Thompson volume and vice versa. The dual of Gromov’s mass* is Gromov’s mass, which however lacks good convexity properties and is used less often. Given a definition of volume µ and a finite-dimensional normed vector space V , n−1 there is an induced (n 1)-density µn−1 : Λ V R. Such a density may be integrated over (n 1)-dimensional− submanifolds in→V . Given a compact− convex set K V , we let ⊂ Aµ(K) := µn−1 Z∂K be the (n 1)-dimensional surface area of K. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Euclidean Structure in Finite Dimensional Normed Spaces
Euclidean structure in ¯nite dimensional normed spaces A.A. Giannopoulos and V.D. Milman¤ Contents 1 Introduction 2 2 Classical inequalities and isotropic positions 4 2.1 Notation . 4 2.2 Classical Inequalities . 5 2.3 Extremal problems and isotropic positions . 11 3 Background from classical convexity 18 3.1 Steiner's formula and Urysohn's inequality . 18 3.2 Geometric inequalities of \hyperbolic" type . 21 3.3 Continuous valuations on compact convex sets . 24 4 Dvoretzky's theorem and concentration of measure 26 4.1 Introduction . 26 4.2 Concentration of measure on the sphere and a proof of Dvoretzky's theorem . 28 4.3 Probabilistic and global form of Dvoretzky's Theorem . 32 4.4 Applications of the concentration of measure on the sphere . 34 4.5 The concentration phenomenon: L¶evyfamilies . 35 4.6 Dvoretzky's theorem and duality . 37 4.7 Isomorphic versions of Dvoretzky's Theorem . 39 5 The Low M ¤-estimate and the Quotient of Subspace Theorem 40 5.1 The Low M ¤-estimate . 40 5.2 The `-position . 42 5.3 The quotient of subspace theorem . 43 5.4 Variants and applications of the Low M ¤-estimate . 44 ¤The authors acknowledge the hospitality of the Erwin SchrÄodingerInternational In- stitute for Mathematical Physics in Vienna, where this work has been completed. The second named author was supported in part by the Israel Science Foundation founded by the Academy of Sciences and Humanities. 1 6 Isomorphic symmetrization and applications to classical convexity 46 6.1 Estimates on covering numbers . 46 6.2 Isomorphic symmetrization and applications to classical convexity . -
Convexity: the Brunn–Minkowski Theory
Chapter 1 Convexity: the Brunn–Minkowski theory 1.1 Basic facts on convex bodies n 2 2 1=2 We work in the Euclidean space (R ; j · j), where jxj = x1 + ··· + xn . We denote by h · ; · i the corresponding inner product. We say that a subset K ⊂ Rn is convex if for every x, y 2 K and λ 2 [0; 1], we have λx + (1 − λ)y 2 K. We say that K ⊂ Rn is a convex body if K is convex, compact, with non-empty interior. It is convenient to define a distance on the set of convex bodies in Rn. First, given n K ⊂ R and " > 0, we denote by K" the "-enlargement of K, defined as n K" = fx 2 R : 9y 2 K; jx − yj 6 "g: In other words, K" is the union of closed balls of radius " with centers in K. The Hausdorff distance between two non-empty compact subsets K, L ⊂ Rn is then defined as δ(K; L) = inff" > 0 : K ⊂ L" and L ⊂ K"g: We check (check!) that δ is a proper distance on the space of non-empty compact subsets of Rn. Some basic but important examples of convex bodies in Rn are n n 1. The unit Euclidean ball, defined as B2 = fx 2 R : jxj 6 1g. n n 2. The (hyper)cube B1 = [−1; 1] . n n 3. The (hyper)octahedron B1 = fx 2 R : jx1j + ··· + jxnj 6 1g. 1 n These examples are unit balls for the `p norm on R for p = 2; 1; 1. -
Contributing Vertices-Based Minkowski Sum Computation of Convex Polyhedra Hichem Barki, Florence Denis, Florent Dupont
Contributing vertices-based Minkowski sum computation of convex polyhedra Hichem Barki, Florence Denis, Florent Dupont To cite this version: Hichem Barki, Florence Denis, Florent Dupont. Contributing vertices-based Minkowski sum computation of convex polyhedra. Computer-Aided Design, Elsevier, 2009, 7, 41, pp.525-538. 10.1016/j.cad.2009.03.008. hal-01437644 HAL Id: hal-01437644 https://hal.archives-ouvertes.fr/hal-01437644 Submitted on 6 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Contributing vertices-based Minkowski sum computation of convex polyhedra Hichem Barki a;∗ Florence Denis a Florent Dupont a aUniversit´ede Lyon, CNRS Universit´eLyon 1, LIRIS, UMR5205 43 Bd. du 11 novembre 1918, F-69622 Villeurbanne, France Abstract Minkowski sum is an important operation. It is used in many domains such as: computer-aided design, robotics, spatial planning, mathematical morphology, and image processing. We propose a novel algorithm, named the Contributing Vertices- based Minkowski Sum (CVMS) algorithm for the computation of the Minkowski sum of convex polyhedra. The CVMS algorithm allows to easily obtain all the facets of the Minkowski sum polyhedron only by examining the contributing vertices|a concept we introduce in this work, for each input facet.