Abstract Let K be a symmetric convex body and K ○ its polar body. Call ϕ ( K ) = 1 | K | | K ○ | ∫ K ∫ K ○ 〈 x , y 〉 2 d y d x . 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.