$L_p$ Radial Minkowski Homomorphisms

Abstract

Intersection bodies define a continuous and $GL(n)$ contravariant valuation which plays a crucial role in the solution of the Busemann-Petty problem. In this paper, we introduce the concept of $L_{p}$ radial Minkowski homomorphisms and consider the Busemann-Petty type problem whether $\Phi_{p} K \subseteq \Phi_{p} L$ implies $V(K) \leq V(L)$, where $\Phi_{p}$ is a homogeneous of degree $\displaystyle \left(\frac{n}{p}-1\right)$, continuous operator on star bodies which is an $SO(n)$ equivariant valuation. Previous results by Schuster are generalized to a large class of $L_{p}$ radial valuations.