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Diversities and the Generalized Circumradius
- Publication Year :
- 2021
-
Abstract
- The generalized circumradius of a set of points $A \subseteq \mathbb{R}^d$ with respect to a convex body $K$ equals the minimum value of $\lambda \geq 0$ such that $A$ is contained in a translate of $\lambda K$. Each choice of $K$ gives a different function on the set of bounded subsets of $\mathbb{R}^d$; 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 generalised circumradius to a finite subset of $\mathbb{R}^d$. We obtain elegant characterizations in the case that $K$ is a simplex or parallelotope.<br />Comment: To be published in Discrete and Computational Geometry
- Subjects :
- Mathematics - Metric Geometry
52A20, 51F99
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2110.13383
- Document Type :
- Working Paper