Extremal convex sets for Sylvester–Busemann type functionals

Abstract

The Sylvester (d+2)-points problem deals with the probability S(K) that d + 2 random points taken from a convex compact subset K of are not the vertices of any convex polytope and asks for which sets S(K) is minimal or maximal. While it is known that ellipsoids are the only minimizers of S(K), the problem of the maximum is still open, unless d = 2. In this article we study generalizations of S(K), which include the Busemann functional – appearing in the formula for the volume of a convex set in terms of the areas of its central sections – and a functional introduced by Bourgain, Meyer, Milman and Pajor in connection with the local theory of Banach spaces. We also show that for these more general functionals ellipsoids are the only minimizers and, in the two-dimensional case, triangles (or parallelograms, in the symmetric case) are maximizers.