$L_p$ functional Busemann-Petty centroid inequality

Abstract

If $K\subset {\mathbb{R}}^n$ is a convex body and ${\Gamma }_{p}K$ is the $p$-centroid body of $K$, the $L_p$ Busemann-Petty centroid inequality states that $\vol(Γ_pK) \geq \vol(K)$, 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 $r$-mixed volume for $1\le r<n$, and establish as a consequence, a functional version of the $L_p$ Busemann-Petty centroid inequality. \keywords{Convex body, Moment body, Busemann-Petty centroid} }

14 pages. Comments are welcome