Brunn-Minkowski Inequality for p-Harmonic Measures

Abstract

We prove a Brunn--Minkowski inequality for a functional associated with p-harmonic measures for $1<p<n+1$, on a class of convex domains satisfying suitable conditions. Under an additional first-variation assumption, we also prove uniqueness up to dilation about the origin for the corresponding Minkowski problem. We also study geometric conditions involving level sets, support functions, and Gaussian curvature for the problem with a fixed inner body.

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