# Quantitative estimates for the relative isoperimetric problem and its gradient flow outside convex bodies in the plane

## TL;DR

This paper establishes explicit quantitative stability, convergence rates, and rigidity results for the relative isoperimetric problem outside convex bodies in the plane, employing a novel flow approach with optimal constants.

## Contribution

It introduces a flow-based method to prove quantitative stability for minimizers in the relative isoperimetric problem, with explicit constants and optimal exponents.

## Key findings

- Lojasiewicz estimates and rigidity for critical points
- Explicit convergence rates for the gradient flow
- Quantitative stability for minimizers with optimal constants

## Abstract

We prove three related quantitative results for the relative isoperimetric problem outside a convex body $\Omega$ in the plane: (1) {\L}ojasiewicz estimates and quantitative rigidity for critical points, (2) rates of convergence for the gradient flow, and (3) quantitative stability for minimizers. These results come with explicit constants and optimal exponents/rates, and hold whenever a simple two-dimensional auxiliary variational problem for circular arcs outside of $\Omega$ is nondegenerate. The proofs are inter-related, and in particular, for the first time in the context of isoperimetric problems, a flow approach is used to prove quantitative stability for minimizers.