Stability of the Blaschke-Santal\'o and the affine isoperimetric inequality

TL;DR

This paper proves stability versions of the Blaschke-Santaló and affine isoperimetric inequalities for convex bodies in higher dimensions, using symmetry reductions and characterizations of ellipsoids.

Contribution

It introduces a new stability analysis for these inequalities, focusing on symmetric convex bodies and their hyperplane sections, extending previous results.

Findings

Stability versions are established for convex bodies with symmetry.

Reduction to symmetric and rotationally symmetric bodies simplifies analysis.

Ellipsoids are characterized by symmetric hyperplane sections in the stability context.

Abstract

A stability version of the Blaschke-Santal\'o inequality and the affine isoperimetric inequality for convex bodies of dimension n>2 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these convex bodies, a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established.