A local uniqueness theorem for minimizers of Petty's conjectured projection inequality

TL;DR

This paper proves that near the unit ball, the only solutions to a specific projection inequality are ellipsoids, using advanced functional analysis techniques.

Contribution

It introduces a local uniqueness theorem for minimizers of Petty's projection inequality, showing only ellipsoids satisfy the condition near the unit ball.

Findings

Only origin-centered ellipsoids solve the equation near the unit ball.

The inverse function theorem on Banach spaces is effectively applied.

The result characterizes local solutions to Petty's conjectured inequality.

Abstract

Employing the inverse function theorem on Banach spaces, we prove that in a $C^{2}(S^{n-1})$-neighborhood of the unit ball, the only solutions of $\Pi^2K=cK$ are origin-centered ellipsoids. Here $K$ is an $n$-dimensional convex body, $\Pi K$ is the projection body of $K$ and $\Pi^2K=\Pi(\Pi K).$