On optimal disc covers and a new characterization of the Steiner center

Given N   points in the plane P1,P2,…,PNP1,P2,…,PN and a location Ω, the union of discs with diameters [ΩPi][ΩPi], i=1,2,…,Ni=1,2,…,N covers the convex hull of the points. The location ΩsΩs 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.