![]() ![]() "Infinite Dimensional Analysis and Topology", dedicated to 70th anniversary of Professor Oleh Lopushansky, (16 - 20 October, 2019), The Josefson-Nissenzweig property for locally convex spaces, «Modern Problems of Geometry and Topology and its Applications», (21 - 23 November 2019), Antonyan, (9 - 12 December 2019), Mazatlan, Mexico. The continuity of Darboux functions between manifolds,Ĭonference on Geometric Topology and Related Topics,ĭedicated to the 65th birthday of Sergey A. ![]() The conference dedicated to the 60th anniversary of the algebra department of Kyiv University, (14 - 17 July, 2020), Kyiv, Ukraine. Topology in Arithmetics: the Golomb and Kirch topologies on N, Inspirations in Real Analysis ( - ), Bedlewo, Poland.Īny isometry between the spheres of absolutely smooth 2-dimensional Banach spaces is linear,Īlgebraic and Geometric Methods of Analysis, (25 - 28 May, 2021), Odesa, Ukraine.Ĭontemporary Mathematics in Kielce, (24 - 27 Feb, 2021), Kielce, Poland.Ī semigroup is finite iff it contains no infinite chains and infinite antichains,ĪAA 100 - Arbeitstagung Allgemeine Algebra, (5 - 7 Feb, 2021), Krakow, Poland.Ī universal countable second-countable space,Īlgebraic and geometric methods of analysis, (26 - 30 May, 2020), Odesa, Ukraine. TOPOSYM 2022 (25-29 July 2022), Prague, Czech Republic.Īuthomatic continuity of measurable homomorphisms between topological groups,Īnalysis (12-), Ivano-Frankivsk, Ukraine. The book can be used in the classroom setting for graduates courses or seminars in convex geometry, geometric and convex combinatorics, and convex analysis and optimization. Selected Topics in Convex Geometry requires of the reader only a basic knowledge of geometry, linear algebra, analysis, topology, and measure theory. Part 3 is devoted to the important class of star bodies and selectors for convex and star bodies, including a presentation of two famous problems of geometric tomography: the Shephard problem and the Busemann-Petty problem. In Part 2 the reader is given a survey on curvature and surface area measures and extensions of the class of convex bodies. The theory of intrinsic volumes for convex bodies, along with the Hadwiger characterization theorems, whose proofs are based on beautiful geometric ideas such as the rounding theorems and the Steiner formula, are treated in Part 1. This exposition, examining in detail those topics in convex geometry that are concerned with Euclidean space, is enriched by numerous examples, illustrations, and exercises, with a good bibliography and index. The field of convex geometry has become a fertile subject of mathematical activity in the past few decades. The orbit space cb(ℝnn)/Aff(n) is homeomorphic to the Banach-Mazur compactum BM(n), while cc(ℝnn)/ O(n) is homeomorphic to the open cone over BM(n). In particular, we show that if K ⊃ O(n) is a closed subgroup that acts nontransitively on the unit sphere ℝSn-1, then the orbit space cc(ℝn)/K is homeomorphic to the Hilbert cube with a point removed, while cb(ℝn)/K is a contractible Q-manifold homeomorphic to the product (E(n)/K) x Q. Furthermore, we investigate the action of the orthogonal group O(n) on cc(ℝn). This is applied to show that cb(ℝn) is homeomorphic to the product Q x ℝn(n 3)/2, where Q stands for the Hilbert cube. ![]() We prove that the space E(n) of all n-dimensional ellipsoids is an Aff (n)-equivariant retract of cb(ℝn). In this paper we discover several fundamental properties of the natural action of the affine group Aff(n) on cb(ℝn). Let cb(ℝn) be the subset of cc(ℝn) consisting of all compact convex bodies. For every tuple $d_1,\dots, d_l\geq 2,$ let $\mathbb.$ We also prove that both the projective and the injective tensor products of $0$-symmetric convex bodies are continuous functions with respect to the Hausdorff distance.įor every n ≥ 2, let cc(ℝn) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space ℝn endowed with the Hausdorff metric topology. ![]()
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