Abstract
Burago et al. (Graduate studies in mathematics, AMS, 2001) considered quotient metric spaces consisting of orbits with respect to some isometry groups. We extend their approach over some semigroups of transformations. We are concerned with quotient semi-metrics in spaces of generalized orbits and give conditions sufficient for these semi-metrics to be metrics. We apply our approach to hyperspaces of compact convex subsets of Euclidean n-space and to that of convex bodies, endowed, respectively, with the Hausdorff metric and with the symmetric difference metric.
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Burago, D., Burago, Yu., Ivanov, S.: A course in metric geometry. In: Graduate Studies in Mathematics. AMS (2001)
Gruber P.M., Lettl G.: Isometries of the space of convex bodies in Euclidean space. Bull. Lond. Math. Soc. 12, 455–462 (1980)
Herburt I., Moszyńska M.: On metric products. Colloq. Math. 92(1), 121–133 (1991)
Herburt, I., Moszyńska, M.: Optimal isometries for a pair of compact convex subsets of Rn. In: Convex and Fractal Geometry. Banach Center Publications, vol. 84, pp. 111–120 (2009)
Liapin, E.S.: Semigroups. Translations of Math. Monographs, vol. 3. AMS (1974)
Moszyńska, M.: Selected Topics in Convex Geometry. Birkhäuser (2006)
Pallaschke, D., Urbański, R.: Pairs of compact convex sets. Kluwer, Dordrecht (2002)
Schneider R.: Isometrien des Raumes der konvexen Körper. Colloq. Math. 33, 219–224 (1975)
Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press (1993)
Volčič, A.: Random symmetrizations of measurable sets. Calc. Var. PDE (2012, to appear)
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Bogdewicz, A., Herburt, I. & Moszyńska, M. Quotient metrics with applications in convex geometry. Beitr Algebra Geom 53, 379–397 (2012). https://doi.org/10.1007/s13366-011-0082-2
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DOI: https://doi.org/10.1007/s13366-011-0082-2
Keywords
- Compact convex sets
- Convex bodies
- Hausdorff metric
- Symmetric difference metric
- Quotient semi-metric
- Quotient metric
- Weakly left (right) divisible semigroup