A generalised isodiametric problem

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.