*Submitted Paper*

**Inserted:** 3 jul 2024

**Last Updated:** 3 jul 2024

**Year:** 2024

**Abstract:**

We study the isoperimetric problem for capillary surfaces with a general contact angle \( \theta \in (0, \pi) \), outside convex infinite cylinders with arbitrary two-dimensional convex section. We prove that the capillary energy of any surface supported on any such convex cylinder is strictly larger than that of a spherical cap with the same volume and the same contact angle on a flat support, unless the surface is itself a spherical cap resting on a facet of the cylinder. In this class of convex sets, our result extends for the first time the well-known Choe-Ghomi-RitorĂ© relative isoperimetric inequality, corresponding to the case \( \theta = \pi/2 \), to general angles.

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