Ball and Spindle Convexity with respect to a Convex Body

TL;DR

This paper introduces and studies two new convexity notions related to a convex body C, exploring their properties, separation, Carathéodory numbers, and stability results in the context of Minkowski spaces.

Contribution

It defines $C$-ball convexity and $C$-spindle convexity, analyzing their fundamental properties and characterizing bodies where these convex sets are finitely generated.

Findings

Characterized convex bodies where $C$-ball convex sets are finitely generated.

Analyzed separation properties and Carathéodory numbers for the new convexity notions.

Established stability results for covering numbers and diametrically maximal sets.

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 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 $C$, 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.