Ball Convex Bodies in Minkowski Spaces

TL;DR

This paper explores the geometric properties of ball convex bodies in normed spaces, including separation theorems, characterizations of strict convexity, and representations of these bodies, extending classical convexity concepts.

Contribution

It introduces new results on minimal representations, union-based representations, and convexity properties of ball convex bodies in Minkowski spaces.

Findings

Derived separation theorems for ball convex sets

Characterized strict convexity via ball convexity properties

Established representation results for ball hulls of sets

Abstract

The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset $S$ of a normed space is called ball convex if it coincides with its ball hull, which is obtained as intersection of all balls (of fixed radius) containing $S$. Ball convex sets are closely related to notions like ball polytopes, complete sets, bodies of constant width, and spindle convexity. We will study geometric properties of ball convex bodies in normed spaces, for example deriving separation theorems, characterizations of strictly convex norms, and an application to complete sets. Our main results refer to minimal representations of ball convex bodies in terms of their ball exposed faces, to representations of ball hulls of sets via unions of ball hulls of finite subsets, and to ball convexity of increasing unions of ball convex bodies.