On Optimal Disc Covers and a New Characterization of the Steiner Center

Abstract

Given N points in the plane $P_1 P_2...P_N$ and a location $\Omega$, the union of discs with diameters $[\Omega P_i], i = 1, 2,...N$ covers the convex hull of the points. The location $\Omega_s$ minimizing the area covered by the union of discs, is shown to be the Steiner center of the convex hull of the points. Similar results for $d$-dimensional Euclidean space are conjectured.