Convex bodies with pinched Mahler volume under the centro-affine normal flows

TL;DR

This paper investigates the long-term behavior of smooth, symmetric convex bodies evolving under centro-affine flows, demonstrating convergence to the unit ball for bodies with near-maximal Mahler volume using stability of the Blaschke-Santaló inequality.

Contribution

It establishes convergence results for convex bodies with nearly maximal Mahler volume under centro-affine flows, utilizing a stability version of a key geometric inequality.

Findings

Rescaled solutions converge to the unit ball in smooth topology.

Regularity of solutions is guaranteed for bodies with almost maximum Mahler volume.

The study links Mahler volume stability to geometric flow convergence.

Abstract

We study the asymptotic behavior of smooth, origin-symmetric, strictly convex bodies under the centro-affine normal flows. By means of a stability version of the Blaschke-Santal\'{o} inequality, we obtain regularity of the solutions provided that initial convex bodies have almost maximum Mahler volume. We prove that suitably rescaled solutions converge sequentially to the unit ball in the $\mathcal{C}^{\infty}$ topology modulo $SL(n+1)$.