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).
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Acknowledgments
The authors wish to thank Maria Moszyńska for careful reading, corrections and valuable suggestions for improvement.
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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Bogdewicz, A., Grzybowski, J. Unique metric segments in the hyperspace over a strictly convex Minkowski space. Beitr Algebra Geom 54, 453–467 (2013). https://doi.org/10.1007/s13366-012-0108-4
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DOI: https://doi.org/10.1007/s13366-012-0108-4