The 𝐿_{𝑝} Brunn-Minkowski inequalities for dual quermassintegrals

Abstract

From a convex geometry viewpoint, we proved the L p L_p Brunn-Minkowski inequalities for q q -th dual quermassintegrals, when p ≥ q p\geq q . Based on these inequalities, we obtain relevant uniqueness results of the ( p , q ) (p,q) -th dual curvature measures (up to a dilation when p = q p=q ). As a special case q = 0 q=0 , we obtain the uniqueness of L p L_p integral curvature measure. Part of these uniqueness results were obtained before from different viewpoints.