Vol. 289, No. 2, 2017

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Vol. 320: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Contacts
 
Submission Guidelines
Submission Form
Policies for Authors
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
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
Authors
Thomas Jahn
Faculty of Mathematics
University of Technology
D-09107 Chemnitz
Germany
Horst Martini
Faculty of Mathematics
University of Technology
D-09107 Chemnitz
Germany
Christian Richter
Institute of Mathematics
Friedrich Schiller University
D-07737 Jena
Germany