Original Paper

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry

, Volume 53, Issue 2, pp 379-397

Open Access This content is freely available online to anyone, anywhere at any time.

Quotient metrics with applications in convex geometry

  • Agnieszka BogdewiczAffiliated withFaculty of Mathematics and Information Science, Warsaw University of Technology
  • , Irmina HerburtAffiliated withFaculty of Mathematics and Information Science, Warsaw University of Technology
  • , Maria MoszyńskaAffiliated withInstitute of Mathematics, University of Warsaw Email author 

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 sets Convex bodies Hausdorff metric Symmetric difference metric Quotient semi-metric Quotient metric Weakly left (right) divisible semigroup

Mathematics Subject Classification (2000)

20M20 54E35 54E40 54E25