An Lp-Functional Busemann–Petty Centroid Inequality

Abstract

For a convex body K⊂Rn ⁠, let ΓpK be its Lp -centroid body. The Lp -Busemann–Petty centroid inequality states that vol(ΓpK)≥vol(K) ⁠, with equality if and only if K is an ellipsoid centered at the origin. In this work, we prove inequalities for a type of functional Lr -mixed volume for 1≤r<n and establish, as a consequence, a functional version of the Lp -Busemann–Petty centroid inequality.