An L p -Functional Busemann–Petty Centroid Inequality

For a convex body $K\subset \mathbb{R}^n$, let $\Gamma _pK$ be its $L_p$-centroid body. The $L_p$-Busemann–Petty centroid inequality states that $\operatorname{vol}(\Gamma _pK) \geq \operatorname{vol}(K)$, 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 $L_r$-mixed volume for $1 \leq r < n$ and establish, as a consequence, a functional version of the $L_p$-Busemann–Petty centroid inequality.