Center and radius of a subset of metric space

TL;DR

This paper introduces the concepts of center, radius, quasi-center, and quasi-radius for subsets of metric spaces, extending classical notions from Euclidean spaces and analyzing their properties and relationships.

Contribution

It defines and explores the properties of center, radius, quasi-center, and quasi-radius in metric spaces, generalizing Euclidean concepts and providing new insights into their relationships.

Findings

The center and radius of the largest open ball in a subset belong to the quasi-center and quasi-radius.

In Euclidean spaces, the largest open ball's center and radius relate directly to the subset's center and radius.

The paper characterizes the behavior of these notions under unions and products of subsets.

Abstract

In this paper, we introduce a notion of the center and radius of a subset A of metric space X. In the Euclidean spaces, this notion can be seen as the extension of the center and radius of open/closed balls. The center and radius of a finite product of subsets of metric spaces, and a finite union of subsets of a metric space are also determined. For any subset A of metric space X, there is a natural question to identify the open balls of X with the largest radius that are entirely contained in A. To answer this question, we introduce a notion of quasi-center and quasi-radius of a subset A of metric space X. We prove that the center of the largest open balls contained in A belongs to the quasi-center of A, and its radius is equal to the quasi-radius of A. In particular, for the Euclidean spaces, we see that the center of largest open balls contained in A belongs to the center of A, and its radius is equal to the radius of A.