Reverse isoperimetric inequality in two-dimensional Alexandrov spaces

TL;DR

This paper establishes a reverse isoperimetric inequality for disc-like domains in two-dimensional Alexandrov spaces with curvature bounds, providing insights into geometric properties and equality cases.

Contribution

It introduces a reverse isoperimetric inequality in Alexandrov spaces, extending classical results to spaces with curvature bounds and analyzing the equality conditions.

Findings

Proved a reverse isoperimetric inequality for disc domains in Alexandrov spaces.

Characterized the equality case in the inequality.

Extended isoperimetric concepts to non-smooth curvature-bounded spaces.

Abstract

We prove a reverse isoperimetric inequality for domains homeomorphic to a disc with the boundary of curvature bounded below lying in two-dimensional Alexandrov spaces of curvature $\geqslant c$. We also study the equality case.