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

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Original Paper

DOI: 10.1007/s13366-012-0108-4

Cite this article as:
Bogdewicz, A. & Grzybowski, J. Beitr Algebra Geom (2013) 54: 453. doi:10.1007/s13366-012-0108-4
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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 
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© The Author(s) 2012

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceWarsaw University of TechnologyWarsawPoland
  2. 2.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznanPoland

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