The weighted isoperimetric inequality and Sobolev inequality outside convex sets

In this paper, we establish a weighted capillary isoperimetric inequality outside convex sets using the ${\lambda }_w$-ABP method. The weight function $w$ is assumed to be positive, even, and homogeneous of degree $\alpha$, such that $w^{1\mathrm{/}\alpha }$ is concave on ${\mathbb{R}}^n$. Based on the weighted isoperimetric inequality, we develop a technique of capillary Schwarz symmetrization outside convex sets, and establish a weighted Pólya-Szegö principle and a sharp weighted capillary Sobolev inequality outside convex domain. Our result can be seen as an extension of the weighted Sobolev inequality in the half-space established by Ciraolo-Figalli-Roncoroni in \cite{CFR}.