Qualitative Properties of $k-$Center Problems

TL;DR

This paper explores generalized k-center problems using Minkowski gauges, establishing properties of optimal solutions and conditions for local optima in higher-dimensional convex settings.

Contribution

It extends classical k-center problems to generalized convex balls and analyzes the properties and existence of optimal and local solutions.

Findings

Fundamental properties of global optimal solutions established.

Sufficient conditions for local optimal solutions provided.

Illustrative examples clarify the generalized problem setting.

Abstract

In this paper, we study generalized versions of the k-center problem, which involves finding k circles of the smallest possible equal radius that cover a finite set of points in the plane. By utilizing the Minkowski gauge function, we extend this problem to generalized balls induced by various convex sets in finite dimensions, rather than limiting it to circles in the plane. First, we establish several fundamental properties of the global optimal solutions to this problem. We then introduce the notion of local optimal solutions and provide a sufficient condition for their existence. We also provide several illustrative examples to clarify the proposed problems.