On a generalized hexagonal dart board problem

Jung's theorem says that planar sets of diameter $1$ can be covered by a closed circular disc of radius 1√3. In this paper we will restrict ourselves to finite point sets and consider a fractional Jung-type problem. Let Pₙ be the family of all finite sets of n points of diameter $1$. Let the function value Nₙ(r) (0 < r ≤ 1) be the largest integer k so that for every point set P ∈ Pₙ there is a circle of radius r which covers at least k points of P. We give many lower and upper bounds for Nₙ(r) and determine intervals of $(0,1]$ where exact values of Nₙ(r) can be determined. Among others we show the n≤ N₃ₙ(1/2) ≤ n+1 and Nₙ(1/4) = n/7 for all n≠ 7.