Topological properties of sets represented by an inequality involving distances

TL;DR

This paper explores the topological properties of Voronoi cells in uniformly convex normed spaces, demonstrating when the interior, boundary, and closure behave as expected, using a classical inequality.

Contribution

It establishes conditions under which the topological properties of Voronoi cells hold in infinite-dimensional uniformly convex spaces, extending previous understanding.

Findings

The interior of a Voronoi cell corresponds to strict inequality regions.

The boundary of a Voronoi cell corresponds to equality regions.

The closure of a Voronoi cell equals the closure of its interior under certain conditions.

Abstract

Consider a set represented by an inequality. An interesting phenomenon which occurs in various settings in mathematics is that the interior of this set is the subset where strict inequality holds, the boundary is the subset where equality holds, and the closure of the set is the closure of its interior. This paper discusses this phenomenon assuming the set is a Voronoi cell induced by given sites (subsets), a geometric object which appears in many fields of science and technology and has diverse applications. Simple counterexamples show that the discussed phenomenon does not hold in general, but it is established in a wide class of cases. More precisely, the setting is a (possibly infinite dimensional) uniformly convex normed space with arbitrary positively separated sites. An important ingredient in the proof is a strong version of the triangle inequality due to Clarkson (1936), an interesting inequality which has been almost totally forgotten.