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

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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).


Convex body Strict convexity Minkowski space Hausdorff metric Metric segment 

Mathematics Subject Classification (2000)

Primary 52A10 52A29 Secondary 52A99 



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

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Copyright information

© 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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