Ball convex bodies in Minkowski spaces

Thomas Jahn, Horst Martini and Christian Richter

Vol. 289 (2017), No. 2, 287–316

DOI: 10.2140/pjm.2017.289.287

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 the 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.

To our teachers, colleagues and friends, Prof. Dr. Johannes Böhm, on the occasion of his 90th birthday, and Prof. Dr. Eike Hertel, on the occasion of his 75th birthday.

Keywords

ball convex body, ball hull, ball polytope, b-exposed point, b-face, Carathéodory's theorem, circumball, complete set, exposed b-face, Minkowski space, normed space, separation theorem, spindle convexity, strictly convex norm, supporting sphere

Mathematical Subject Classification 2010

Primary: 46B20, 52A01, 52A20, 52A21, 52A35

Milestones

Received: 11 March 2016

Revised: 3 January 2017

Accepted: 9 January 2017

Published: 19 June 2017