Sharp reverse isoperimetric inequalities in nonpositively curved cones

TL;DR

This paper establishes sharp reverse isoperimetric inequalities for domains in nonpositively curved cones, identifying minimal area configurations for disks and triangles with fixed perimeter and angles.

Contribution

It introduces new sharp reverse isoperimetric inequalities in nonpositively curved cones, extending classical results to these geometric settings.

Findings

Disks in Euclidean cones with angle ≥ 2π have minimal area for given perimeter and curvature.

Triangles in Euclidean and hyperbolic cones with angle ≥ 2π have minimal area for fixed side lengths and angles.

Results are sharp and optimal for the specified geometric conditions.

Abstract

We prove a pair of sharp reverse isoperimetric inequalities for domains in nonpositively curved surfaces: (1) metric disks centered at the vertex of a Euclidean cone of angle at least $2\pi$ have minimal area among all nonpositively curved disks of the same perimeter and the same total curvature; (2) geodesic triangles in a Euclidean (resp. hyperbolic) cone of angle at least $2\pi$ have minimal area among all nonpositively curved geodesic triangles (resp. all geodesic triangles of curvature at most $-1$) with the same side lengths and angles.