Uniqueness of circumcenters in generalized Minkowski spaces

TL;DR

This paper explores the conditions under which bounded sets in generalized Minkowski spaces have unique circumballs, extending classical results to non-symmetric convex bodies and analyzing the geometry of circumcenter sets.

Contribution

It establishes new equivalences for the uniqueness of circumballs in non-symmetric convex bodies and characterizes the dimensions of circumcenter sets in generalized Minkowski spaces.

Findings

Unique circumball for all bounded sets iff all sets of size ≤ n have unique circumballs.

Characterization of the geometry of the unit ball related to circumball uniqueness.

Bound on the dimension of the set of all circumcenters for arbitrary bounded sets.

Abstract

In an $n$-dimensional normed space every bounded set has a unique circumball if and only if every set of cardinality two has a unique circumball and if and only if the unit ball of the space is strictly convex. When the symmetry of the norm is dropped, i.e., when the centrally symmetric unit ball is replaced by an arbitrary convex body, then the above three conditions are no longer equivalent. We show for the latter case that every bounded set has a unique circumball if and only if every set of cardinality at most $n$ has a unique circumball. We also give an equivalent condition in terms of the geometry of the unit ball. In similar terms we answer the following more general question for every $k \in \{0,\ldots,n-2\}$: When are the dimensions of the sets of all circumcenters of arbitrary bounded sets not larger than $k$?