On the stability of the $p$-affine isoperimetric inequality

TL;DR

This paper proves a stability version of the $p$-affine isoperimetric inequality for origin-symmetric convex bodies in the plane, showing near equality implies the shape is close to an ellipse after transformation.

Contribution

It introduces a stability result for the $p$-affine isoperimetric inequality using affine normal flow in $ eal^2$, extending understanding of equality cases.

Findings

Near equality in the inequality implies the shape is close to an ellipse.

The stability is quantified in the Hausdorff distance after a special linear transformation.

The result applies to convex bodies with the same area as an ellipse.

Abstract

Employing the affine normal flow, we prove a stability version of the $p$-affine isoperimetric inequality for $p\geq1$ in $\mathbb{R}^2$ in the class of origin-symmetric convex bodies. That is, if $K$ is an origin-symmetric convex body in $\mathbb{R}^2$ such that it has area $\pi$ and its $p$-affine perimeter is close enough to the one of an ellipse with the same area, then, after applying a special linear transformation, $K$ is close to an ellipse in the Hausdorff distance.