Volumes of projection bodies

Abstract

C. M. Petty has conjectured the minimum value for a certain affine-invariant functional denned on the class of convex bodies. We give sharp bounds for this functional on a certain subclass of convex bodies, and we give a counterexample to an upper bound proposed by R. Schneider for the class of centrally symmetric convex bodies. We conjecture that the simplex provides the maximum on the class of all convex bodies, while the largest centrally symmetric subset of a simplex gives a sharp upper bound on the class of all centrally symmetric convex bodies.