6 research outputs found
On the Hadwiger numbers of starlike disks
The Hadwiger number of a topological disk in is the
maximal number of pairwise nonoverlapping translates of that touch . It
is well known that for a convex disk, this number is six or eight. A conjecture
of A. Bezdek., K. and W. Kuperberg says that the Hadwiger number of a starlike
disk is at most eight. A. Bezdek proved that this number is at most seventy
five for any starlike disk. In this note, we prove that the Hadwiger number of
a starlike disk is at most thirty five. Furthermore, we show that the Hadwiger
number of a topological disk such that (\conv J) \setminus J is
connected, is six or eight.Comment: 13 pages, 8 figure
Ball and Spindle Convexity with respect to a Convex Body
Let be a convex body. We introduce two notions of
convexity associated to C. A set is -ball convex if it is the
intersection of translates of , or it is either , or . The -ball convex hull of two points is called a -spindle. is
-spindle convex if it contains the -spindle of any pair of its points. We
investigate how some fundamental properties of conventional convex sets can be
adapted to -spindle convex and -ball convex sets. We study separation
properties and Carath\'eodory numbers of these two convexity structures. We
investigate the basic properties of arc-distance, a quantity defined by a
centrally symmetric planar disc , which is the length of an arc of a
translate of , measured in the -norm, that connects two points. Then we
characterize those -dimensional convex bodies for which every -ball
convex set is the -ball convex hull of finitely many points. Finally, we
obtain a stability result concerning covering numbers of some -ball convex
sets, and diametrically maximal sets in -dimensional Minkowski spaces.Comment: 27 pages, 5 figure
On a normed version of a Rogers-Shephard type problem
A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of n-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities for the four types of volumes regularly used in the literature: Busemann, Holmes-Thompson, and Gromov's mass and mass*. We examine the problem also for higher dimensions, and for centrally symmetric convex bodies