Search Results
You are looking at 1 - 10 of 13 items for
- Author or Editor: Käroly Bezdek x
- Refine by Access: All Content x
Summary
The Illumination Conjecture was raised independently by Boltyanski and
Hadwiger in 1960. According to this conjecture any
Abstract
Let S d-1 denote the (d − 1)-dimensional unit sphere centered at the origin of the d-dimensional Euclidean space. Let 0 < α < π. A set P of points in S d-1 is called almost α-equidistant if among any three points of P there is at least one pair lying at spherical distance α. In this note we prove upper bounds on the cardinality of P depending only on d.
Abstract
Abstract
In this paper we prove some stronger versions of Danzer-Grnbaum's theorem including the following stability-type result. For 0 < α < 14π/27 the maximum number of vertices of a convex polyhedron in E 3 such that all angles between adjacent edges are bounded from above by α is 8. One of the main tools is the spherical geometry version of Pl's theorem.
Abstract
Let 𝔼 𝑑 denote the 𝑑-dimensional Euclidean space. The 𝑟-ball body generated by a given set in 𝔼 𝑑 is the intersection of balls of radius 𝑟 centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke–Santaló-type inequalities for 𝑟-ball bodies: for all 0 < 𝑘 < 𝑑 and for any set of given 𝑑-dimensional volume in 𝔼 𝑑 the 𝑘-th intrinsic volume of the 𝑟-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.