Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry

, Volume 53, Issue 2, pp 379–397

Quotient metrics with applications in convex geometry

  • Agnieszka Bogdewicz
  • Irmina Herburt
  • Maria Moszyńska
Open AccessOriginal Paper

DOI: 10.1007/s13366-011-0082-2

Cite this article as:
Bogdewicz, A., Herburt, I. & Moszyńska, M. Beitr Algebra Geom (2012) 53: 379. doi:10.1007/s13366-011-0082-2

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.

Keywords

Compact convex setsConvex bodiesHausdorff metricSymmetric difference metricQuotient semi-metricQuotient metricWeakly left (right) divisible semigroup

Mathematics Subject Classification (2000)

20M2054E3554E4054E25
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Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Agnieszka Bogdewicz
    • 1
  • Irmina Herburt
    • 1
  • Maria Moszyńska
    • 2
  1. 1.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland
  2. 2.Institute of MathematicsUniversity of WarsawWarsawPoland