The unit ball is an attractor of the intersection body operator

Abstract

The intersection body of a ball is again a ball. So, the unit ball B d ⊂ 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. E. Lutwak asked if there is any other star-shaped body that satisfies this property. 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 . We will also discuss a harmonic analysis version of this question, which studies the Radon transforms of powers of a given function.