Ball and Spindle Convexity with respect to a Convex Body
Abstract
Let $C\subset {\mathbb R}^n$ be a convex body. We introduce two notions of convexity associated to C. A set $K$ is $C$ball convex if it is the intersection of translates of $C$, or it is either $\emptyset$, or ${\mathbb R}^n$. The $C$ball convex hull of two points is called a $C$spindle. $K$ is $C$spindle convex if it contains the $C$spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to $C$spindle convex and $C$ball convex sets. We study separation properties and Carathéodory numbers of these two convexity structures. We investigate the basic properties of arcdistance, a quantity defined by a centrally symmetric planar disc $C$, which is the length of an arc of a translate of $C$, measured in the $C$norm, that connects two points. Then we characterize those $n$dimensional convex bodies $C$ for which every $C$ball convex set is the $C$ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some $C$ball convex sets, and diametrically maximal sets in $n$dimensional Minkowski spaces.
 Publication:

arXiv eprints
 Pub Date:
 October 2011
 arXiv:
 arXiv:1110.4823
 Bibcode:
 2011arXiv1110.4823L
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Combinatorics;
 52A30;
 52A35;
 52C17
 EPrint:
 27 pages, 5 figures