On the Sobolev distance of convex bodies

Summary LetK^d denote the cone of all convex bodies in the Euclidean spaceK^d. The mappingK h_K of each bodyK K^d onto its support function induces a metric_w onK^d by"_w(K, L)h_L–h_Kw where _w is the Sobolev I-norm on the unit sphere. We call_w(K, L) the Sobolev distance ofK andL. The goal of our paper is to develop some fundamental properties of the Sobolev distance.