Minsum hyperspheres in normed spaces

Abstract

We study the minsum hypersphere problem in finite dimensional real Banach spaces: given a finite set D of (positively weighted) points in an n-dimensional normed space (n≥2), find a minsum hypersphere, i.e., a homothet of the unit sphere of this space that minimizes the sum of (weighted) distances between the hypersphere and the points of D.We show existence results of the following type: there are situations where minsum hyperspheres do not exist, no point-shaped hypersphere can be optimal, and for any norm there exists a set of points D such that a hyperplane is better than any proper hypersphere. We also prove that the intersection of a minsum hypersphere S and conv(D) is non-empty, that D⊆conv(S) implies |S∩conv(D)|≥2, and that |S∩conv(D)|<∞ implies S∩conv(D)⊆D. A certain halving criterion regarding the sums of weights inside and outside of S is verified, and various further results are obtained for large classes of norms, like strictly convex, smooth, and polyhedral norms.