Maximal r-Diameter Sets and Solids of Constant Width

TL;DR

This paper reviews the theory of r-maximal sets in metric spaces, focusing on Euclidean solids of constant diameter, and introduces a simple construction method for generating many such solids.

Contribution

It provides a comprehensive review and a new simple construction method for solids of constant diameter in Euclidean spaces.

Findings

Established the equivalence between r-maximal sets and solids of constant diameter in Euclidean space.

Presented a novel, straightforward construction technique for generating a wide class of these solids.

Abstract

We recall the definition of an r-maximal set in a metric space as a maximal subset of diameter r. In the special case when the metric space is Euclidean such a set is exactly a solid of constant diameter r. In the process of reviewing the theory of these objects we provide a simple construction which generates a large class of such solids.