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

2010/03/22 by Alexander Fish, Fish, A., Fëdor Nazarov +5
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1003.4056

openalex publication_date 2010/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.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.

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