Iterations of the projection body operator and a remark on Petty's conjectured projection inequality

TL;DR

This paper demonstrates that convex bodies with nearly uniform surface area measures converge to a sphere under repeated projection body operations, providing insights into Petty's projection inequality and related geometric inequalities.

Contribution

It establishes convergence of iterated projection bodies to the sphere for bodies with smooth, nearly constant surface measures, and offers new local solutions to Petty's conjecture.

Findings

Convex bodies with smooth surface measures close to constant converge to the sphere under iteration.

Ellipsoids are shown to be local solutions to Petty's conjectured inequality.

Improves existing inequalities related to projection bodies.

Abstract

We prove that if a convex body has absolutely continuous surface area measure, whose density is sufficiently close to the constant, then the sequence $\{\Pi^mK\}$ of convex bodies converges to the ball with respect to the Banach-Mazur distance, as $m\rightarrow\infty$. Here, $\Pi$ denotes the projection body operator. Our result allows us to show that the ellipsoid is a local solution to the conjectured inequality of Petty and to improve a related inequality of Lutwak.