Diversities and the generalized circumradius

David Bryant, Katharina T. Huber, Vincent Moulton, Paul F. Tupper

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The generalized circumradius of a set of points A⊆ R d with respect to a convex body K equals the minimum value of λ≥ 0 such that a translate of λK contains A. Each choice of K gives a different function on the set of bounded subsets of R d; we characterize which functions can arise in this way. Our characterization draws on the theory of diversities, a recently introduced generalization of metrics from functions on pairs to functions on finite subsets. We additionally investigate functions which arise by restricting the generalized circumradius to a finite subset of R d. We obtain elegant characterizations in the case that K is a simplex or parallelotope.

Original languageEnglish
Pages (from-to)1862–1883
Number of pages22
JournalDiscrete and Computational Geometry
Issue number4
Early online date13 Apr 2023
Publication statusPublished - Dec 2023


  • Convex geometry
  • Diversity
  • Generalized Minkowski spaces
  • Generalized circumradius
  • Metric geometry

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