$L_p$ Brunn-Minkowski inequality for weighted dual quermassintegrals

Abstract

We investigate the $L_p$ Brunn-Minkowski inequality for dual quermassintegrals in weighted measure spaces, which is a special class of rotationally invariant measures proposed by Cordero-Erausquin and Rotem [Ann. Probab., {\bf 51} (2023)]. Specifically, the weighted dual quermassintegral is defined by integrating the radial density $|x|^{q-n}ϕ(|x|)$ for $q\in(0,n]$, where $ϕ$ is a positive radially non-increasing weight, it recovers the classical dual quermassintegral when $ϕ\equiv1$. For $p\geq1$, we prove the $L_p$ Brunn-Minkowski inequality with concavity exponent $p/q$ under the condition that $t\mapsto\logϕ(e^t)$ is concave, which is exactly the natural convexity condition from Cordero-Erausquin and Rotem's paper in general, improving the exponent $1/n$ when $p=1$. For $p\in(0,1)$, we obtain the result with exponent $p/q$ under more strictly weight assumptions, together with explicit lower bounds for the admissible range of $p$.