Fixed points of Minkowski valuations

TL;DR

This paper proves that under certain regularity conditions, the only fixed points of a class of Minkowski valuations near the unit ball are balls themselves, offering new insights into geometric inequalities.

Contribution

It generalizes previous results by Ivaki on projection bodies and introduces a novel approach to Petty's conjectured projection inequality using class reduction techniques.

Findings

Balls are the only solutions to the fixed-point problem near the unit ball.

The results extend Ivaki's work on projection bodies.

Suggests a new approach to Petty's conjecture.

Abstract

It is shown that for any sufficiently regular even Minkowski valuation $\Phi$ which is homogeneous and intertwines rigid motions, there exists a neighborhood of the unit ball, where balls are the only solutions to the fixed-point problem $\Phi^2 K = \alpha K$. This significantly generalizes results by Ivaki for projection bodies and suggests, via the Lutwak--Schneider class reduction technique, a new approach to Petty's conjectured projection inequality.