Minsum hyperspheres in normed spaces

We study the minsum hypersphere problem   in finite dimensional real Banach spaces: given a finite set DD of (positively weighted) points in an nn-dimensional normed space (n≥2)(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 DD.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 DD such that a hyperplane is better than any proper hypersphere. We also prove that the intersection of a minsum hypersphere SS 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 SS is verified, and various further results are obtained for large classes of norms, like strictly convex, smooth, and polyhedral norms.