Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry

, Volume 54, Issue 1, pp 453–467

Unique metric segments in the hyperspace over a strictly convex Minkowski space

Authors

    • Faculty of Mathematics and Computer ScienceWarsaw University of Technology
  • Jerzy Grzybowski
    • Faculty of Mathematics and Computer ScienceAdam Mickiewicz University

DOI: 10.1007/s13366-012-0108-4

Abstract

Let \({(\mathbb{R}^{n}, \| \cdot \|_{\mathbb{B}})}\) be a Minkowski space (finite dimensional Banach space) with the unit ball \({\mathbb{B}}\) , and let \({\varrho_H^{\mathbb{B}}}\) be the Hausdorff metric induced by \({\|\cdot\|_{\mathbb{B}}}\) in the hyperspace \({\mathcal{K}^{n}}\) of convex bodies (compact, convex subsets of \({\mathbb{R}^{n}}\) with nonempty interior). Schneider (Bull. Soc. Roy. Sci. Li‘ege 50:5–7, 1981) characterized pairs of elements of \({\mathcal{K}^{n}}\) which can be joined by unique metric segments with respect to \({\varrho_H}\) —the Hausdorff metric induced by the Euclidean norm \({\|\cdot \|_{{\rm B}^{n}}}\) . In Bogdewicz and Grzybowski (Banach Center Publ., Warsaw, 75–88, 2009) we proved a counterpart of Schneider’s theorem for the hyperspace \({(\mathcal{K}^{2},\varrho_H^{\mathbb{B}})}\) over any two-dimensional Minkowski space. In this paper we characterize pairs of convex bodies in \({\mathcal{K}^{n}}\) which can be joined by unique metric segments with respect to \({\varrho_H^{\mathbb{B}}}\) for a strictly convex unit ball \({\mathbb{B}}\) and an arbitrary dimension n (Theorem 3.1).

Keywords

Convex body Strict convexity Minkowski space Hausdorff metric Metric segment

Mathematics Subject Classification (2000)

Primary 52A10 52A29 Secondary 52A99

Acknowledgments

The authors wish to thank Maria Moszyńska for careful reading, corrections and valuable suggestions for improvement.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Copyright information

© The Author(s) 2012