## Abstract

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 language | English |
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Pages (from-to) | 1862–1883 |

Number of pages | 22 |

Journal | Discrete and Computational Geometry |

Volume | 70 |

Issue number | 4 |

Early online date | 13 Apr 2023 |

DOIs | |

Publication status | Published - Dec 2023 |

## Keywords

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