A New Affine Invariant for Polytopes and Schneider’s Projection Problem

New affine invariant functionals for convex polytopes are introduced. Some sharp affine isoperimetric inequalities are established for the new functionals. These new inequalities lead to fairly strong volume estimates for projection bodies. Two of the new affine isoperimetric inequalities are extensions of Ball’s reverse isoperimetric inequalities. If K is a convex body (i.e., a compact, convex subset with nonempty interior) in Euclidean n-space, R, then on the unit sphere, Sn−1, its support function, h(K, · ) : Sn−1 → R, is defined for u ∈ Sn−1 by h(K, u) = max{u · y : y ∈ K}, where u ·y denotes the standard inner product of u and y. The projection body, ΠK, of K can be defined as the convex body whose support function, for u ∈ Sn−1, is given by h(ΠK,u) = voln−1(K|u⊥), where voln−1 denotes (n − 1)-dimensional volume and K|u⊥ denotes the image of the orthogonal projection of K onto the codimension 1 subspace orthogonal to u. An important unsolved problem regarding projection bodies is Schneider’s projection problem: What is the least upper bound, as K ranges over the class of origin-symmetric convex bodies in R, of the affine-invariant ratio (∗) [V (ΠK)/V (K)n−1]1/n, 1991 Mathematics Subject Classification. 52A40.