Total positive curvature and the equality case in the relative isoperimetric inequality outside convex domains

TL;DR

This paper proves the precise conditions under which the relative isoperimetric inequality outside convex sets achieves equality, extending understanding of geometric inequalities in convex analysis.

Contribution

It establishes the equality case for the relative isoperimetric inequality outside any convex set with non-empty interior, a previously unresolved problem.

Findings

Equality cases characterized for arbitrary convex sets.

Extension of isoperimetric inequality understanding outside convex domains.

Provides a complete solution to the equality case in this context.

Abstract

We settle the case of equality for the relative isoperimetric inequality outside any arbitrary convex set with not empty interior.