On a characterization of circles and spheres.
On a probability inequality for multivariate normal distribution
On a question of Milman and Alesker concerning the monotonicity of a domain functional.
On a Spherical Integral Transformation and Sections of Star Bodies.
On affine subspaces that illuminate a convex set.
On Antipodal and Adjoint Pairs of Points for Two Convex Bodies.
On Averaging Multisets.
On Bárány's theorems of Carathéodory and Helly type
The paper begins with a self-contained and short development of Bárány’s theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads as follows:...
On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space
Let be a Euclidean space or the Hilbert space ℓ², let k ∈ ℕ with k < dim , and let B be convex and closed in . Let be a collection of linear k-subspaces of . A set C ⊂ is called a -imitation of B if B and C have identical orthogonal projections along every P ∈ . An extremal point of B with respect to the projections under is a point that all closed subsets of B that are -imitations of B have in common. A point x of B is called exposed by if there is a P ∈ such that (x+P) ∩ B = x. In the present...
On Hadwiger's problem on inner parallel bodies
We consider the problem of classifying the convex bodies in the 3-dimensional space depending on the differentiability of their associated quermassintegrals with respect to the one-parameter-depending family given by the inner/outer parallel bodies. It turns out that this problem is closely related to some behavior of the roots of the 3-dimensional Steiner polynomial.
On maximal ot-subsets of the Euclidean plane.
On Multiple Space Subdivisions by Zonotopes.
On point classification in convex sets.
On point sets fixing a convex body from within.
On shadow boundaries of centrally symmetric convex bodies.
On some classes of curves in a projective space
On some vector balancing problems
Let V be an origin-symmetric convex body in , n≥ 2, of Gaussian measure . It is proved that for every choice of vectors in the Euclidean unit ball , there exist signs with . The method used can be modified to give simple proofs of several related results of J. Spencer and E. D. Gluskin.
On splitting infinite-fold covers
Let X be a set, κ be a cardinal number and let ℋ be a family of subsets of X which covers each x ∈ X at least κ-fold. What assumptions can ensure that ℋ can be decomposed into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover ℋ: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of ℝⁿ by polyhedra and by arbitrary convex sets. We focus on...
On star duality of mixed intersection bodies.
On support functions of Compact Convex Sets.