A generalised isodiametric problem

Fix positive integers a and b such that a > b ≥ 2 and a positive real δ > 0. Let S be a planar set of diameter δ 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.