The isoperimetric inequality for the capillary energy outside convex sets

We study the isoperimetric problem for capillary hypersurfaces with a general contact angle $\theta \in (0, \pi)$, outside arbitrary convex sets. We prove that the capillary energy of any surface supported on any such convex set is larger than that of a spherical cap with the same volume and the same contact angle on a flat support, and we characterize the equality cases. This provides a complete solution to the isoperimetric problem for capillary surfaces outside convex sets at arbitrary contact angles, generalizing the well-known Choe-Ghomi-Ritor\'e inequality, which corresponds to the case $\theta=\frac\pi2$.