TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 5, Pages 1767–1779 S 0002-9947(01)02726-X Article electronically published on January 5, 2001 A NEW AFFINE INVARIANT FOR POLYTOPES AND SCHNEIDER’S PROJECTION PROBLEM ERWIN LUTWAK, DEANE YANG, AND GAOYONG ZHANG Abstract. New affine invariant functionals for convex polytopes are intro- duced. 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 exten- sions 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 n , then on the unit sphere, S n−1 , its support function, h(K, · ): S n−1 → R, is defined for u ∈ S n−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 ∈ S n−1 , is given by h(ΠK, u) = vol n−1 (K|u ⊥ ), where vol n−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 pro- jection problem: What is the least upper bound, as K ranges over the class of origin-symmetric convex bodies in R n , of the affine-invariant ratio [V (ΠK)/V (K) n−1 ] 1/n , (∗) where V is used to abbreviate vol n . See [S1], [S2], [SW] and [Le]. Schneider [S1] conjectured that this ratio is maximized by parallelotopes. In [S1], Schneider also presented applications of such results in stochastic geometry. However, a coun- terexample was produced in [Br] to show that this is not the case. We will present a modified version of Schneider’s conjecture that has an affirma- tive answer. In addition, we will obtain an inequality that gives an upper bound for the affine ratio (∗). While our upper bound is not sharp for any n, nevertheless it is asymptotically optimal. To be more specific, in this paper, we introduce a new centro-affine functional U , defined on the class of polytopes, which is closely related to the volume functional V . While in general U (K) <V (K), if K is a random polytope (with many faces), then U (K) is very close to V (K). We shall prove the following variation of Schneider’s projection conjecture: Received by the editors February 26, 2000. 1991 Mathematics Subject Classification. Primary 52A40. Key words and phrases. Affine isoperimetric inequalities, reverse isoperimetric inequalities, projection bodies, asymptotic inequalities. Research supported, in part, by NSF Grant DMS–9803261. c 2001 American Mathematical Society 1767 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use