Fixed and Periodic Points of the Intersection Body Operator

2024/08/15 by Emanuel Milman, Milman, Emanuel, Shahar Shabelman +3 · 2 citations
Mathematics · #Algebraic and Geometric Analysis #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2408.08171

openalex publication_date 2024/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

The intersection body IK of a star-body K in ℝⁿ was introduced by E. Lutwak following the work of H. Busemann, and plays a central role in the dual Brunn-Minkowski theory. We show that when n ≥ 3, I² K = c K iff K is a centered ellipsoid, and hence I K = c K iff K is a centered Euclidean ball, answering long-standing questions by Lutwak, Gardner, and Fish-Nazarov-Ryabogin-Zvavitch. To this end, we recast the iterated intersection body equation as an Euler-Lagrange equation for a certain volume functional under radial perturbations, derive new formulas for the volume of I K, and introduce a continuous version of Steiner symmetrization for Lipschitz star-bodies, which (surprisingly) yields a useful radial perturbation exactly when n≥ 3.

Cited by

• Fixed and periodic points of the intersection body operators of lower orders
• On the homothety conjecture for the body of flotation and the body of buoyancy on a plane

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