On an extension of the Blaschke–Santaló inequality and the hyperplane conjecture

Let K be a symmetric convex body and K ◦ its polar body. Call φ(K) = 1 |K||K ◦| K K ◦ x,y 2 dy dx. It is conjectured that φ(K) is maximum when K is an ellipsoid. In particular this statement implies the Blaschke–Santaló inequality and the hyperplane conjecture. We verify this conjecture when K is restricted to be a p-ball. © 2008 Elsevier Inc. All rights reserved.