Fixed and periodic points of the intersection body operator

• 成果类型: Article
• 署名作者: Milman, Emanuel; Shabelman, Shahar; Yehudayoff, Amir
• 署名单位: Technion Israel Institute of Technology; University of Copenhagen
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01342-z
• 发表日期: 2025
• 页码: 509-558
• 关键词: busemann-petty problem; minkowski-firey theory; convex; Affine; inequalities; Transforms; bodies

摘要

The intersection body I K of a star body K in R-n 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, (IK)-K-2=cK 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. An equivalent formulation of the latter in terms of non-linear harmonic analysis states that a non-negative rho is an element of L-infinity(Sn-1) satisfies R(rho(n-1))=c rho for some c > 0 iff rho is constant, where R denotes the spherical Radon transform. Our proof is entirely geometrical: 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 IK, and introduce a continuous version of Steiner symmetrization for Lipschitz star bodies, which (surprisingly) yields a useful radial perturbation exactly when n >= 3.