The behavior of iterations of the intersection body operator in a small neighborhood of the unit ball

TL;DR

This paper proves that in a small neighborhood around the unit ball, the intersection body operator's iterations converge to the ball, making it a local attractor with no other fixed or periodic points nearby.

Contribution

It establishes the local stability of the unit ball as a fixed point of the intersection body operator in the space of star-shaped bodies.

Findings

Iterations converge to the unit ball near it in Banach-Mazur distance.

The unit ball is a local attractor for the intersection body operator.

No other fixed or periodic points exist close to the unit ball.

Abstract

The intersection body of a ball is again a ball. So, the unit ball $B_d \subset \R^d$ is a fixed point of the intersection body operator acting on the space of all star-shaped origin symmetric bodies endowed with the Banach-Mazur distance.We show that this fixed point is a local attractor, i.e., that the iterations of the intersection body operator applied to any star-shaped origin symmetric body sufficiently close to $B_d$ in Banach-Mazur distance converge to $B_d$ in Banach-Mazur distance. In particular, it follows that the intersection body operator has no other fixed or periodic points in a small neighborhood of $B_d$.