$L_p$ Brunn-Minkowski inequality for weighted dual quermassintegrals

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 $\mathrm{\mid }x{\mathrm{\mid }}^{q-n}\phi (\mathrm{\mid }x\mathrm{\mid })$ for $q\in (0,n]$, where $\phi$ is a positive radially non-increasing weight, it recovers the classical dual quermassintegral when $\phi \equiv 1$. For $p\ge 1$, we prove the $L_p$ Brunn-Minkowski inequality with concavity exponent $p\mathrm{/}q$ under the condition that $t\mapsto \log \phi (e^t)$ is concave, which is exactly the natural convexity condition from Cordero-Erausquin and Rotem’s paper in general, improving the exponent $1\mathrm{/}n$ when $p=1$. For $p\in (0,1)$, we obtain the result with exponent $p\mathrm{/}q$ under more strictly weight assumptions, together with explicit lower bounds for the admissible range of $p$.