Nonlocal energies of convex bodies and their log-Minkowski problems

Abstract

Under a pioneer work in integral geometry, Lutwak, Xi, Yang, and Zhang [19] established recently the variational formula for chord integral Iq(K), for q>0, and defined the q-th chord measure. Meanwhile, they provided sufficient and necessary conditions for the existence of a solution to the chord Minkowski problem, for q>0, and sufficient conditions to solve the symmetric case of the chord log-Minkowski problem when 1≤q≤n+1. It is well known that for q>1, Iq(K) is the Riesz potential of characteristic function of convex body K. In the case of 0<q<1, we discover the new relationship between nonlocal energy and chord integral Iq(K), which gives the representation formula of Iq(K) in the sense of Riesz potential, and we also give a new definition of chord measure by making use of its representation formula of Riesz potentials for 0<q<1. Finally, we solve the symmetric case of chord log-Minkowski problem under a sufficient condition for 0<q<1.