A generalised isodiametric problem

TL;DR

This paper explores a generalized isodiametric problem involving planar sets with diameter constraints and specific distance properties among points, providing bounds and extremal examples based on a Jung-type theorem.

Contribution

It introduces a generalized formulation of the isodiametric problem, extends Jung's theorem, and offers bounds and extremal set constructions for the problem.

Findings

Provided upper bounds on the Lebesgue measure of the set

Extended Jung's theorem to a more general setting

Identified extremal sets in specific cases

Abstract

Fix positive integers $a$ and $b$ such that $a> b\geq 2$ and a positive real $\delta>0$. Let $S$ be a planar set of diameter $\delta$ having the following property: for every $a$ points in $S$, at least $b$ of them have pairwise distances that are all less than or equal to $2$. What is the maximum Lebesgue measure of $S$? In this paper we investigate this problem. We discuss the, devious, motivation that leads to its formulation and provide upper bounds on the Lebesgue measure of $S$. Our main result is based on a generalisation of a theorem that is due to Heinrich Jung. In certain instances we are able to find the extremal set but the general case seems elusive.