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

Let K   be a symmetric convex body and K○K○ its polar body. Callϕ(K)=1|K||K○|∫K∫K○〈x,y〉2dydx. It is conjectured that ϕ(K)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.