On affine subspaces that illuminate a convex set.
On an Ordering for Probability Measures and a Corresponding Integral Representation.
On Antipodal and Adjoint Pairs of Points for Two Convex Bodies.
On area and side lengths of triangles in normed planes
Let be a d-dimensional normed space with norm ||·|| and let B be the unit ball in . Let us fix a Lebesgue measure in with . This measure will play the role of the volume in . We consider an arbitrary simplex T in with prescribed edge lengths. For the case d = 2, sharp upper and lower bounds of are determined. For d ≥ 3 it is noticed that the tight lower bound of is zero.
On areas and integral geometry in Minkowski spaces.
On Asymptotic Volume of Tori.
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 Barbier's theorem
On billiard arcs
On boundaries of unions of sets with positive reach.
On boundary arcs joining antipodal points of a planar convex body.
On braxtopes, a class of generalized simplices.
On Buffon's problem for a lattice and its deformations.
On Busemann surface area of the unit ball in Minkowski spaces.
On Carathéodory's and Krein-Milman's theorems in fully ordered groups
On certain random polygons of large areas.
On closed sets with convex projections in Hilbert space
Let k be a fixed natural number. We show that if C is a closed and nonconvex set in Hilbert space such that the closures of the projections onto all k-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of Hilbert space. In order to prove this result we introduce for convex closed sets B the set consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that is precisely the intersection of all k-imitations...
On complex -predual spaces.
On Convex Bodies that Permit Packings of High Density.