Fixed and periodic points of the intersection body operators of lower orders

2025/10/30 by Lin, Cheng, Xiong, Ge
#42B15 #52A30 #52A38 #52A40 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2510.26381

Abstract

For the intersection body operator of lower order I_iK of a star body K in ℝⁿ, i∈\1, 2,…, n-2\, we prove that I_i²K = cK iff K is an origin-symmetric ball, and hence I_iK = cK iff K is an origin-symmetric ball. Combining the recent breakthrough (case i = n-1) of Milman, Shabelman and Yehudayoff (Invent. Math., 241 (2025), 509-558), slight modifications of two long-standing questions 8.6 and 8.7 posed by R. Gardner (Page 302, Geometric Tomography, Cambridge University Press, 1995) are completely solved. As applications, we show that for the spherical Radon transform R, a non-negative ρ∈ L^∞(S^n-1) satisfies R(ρⁱ) = cρ for some c>0 iff ρ is constant. Also, the generalized Busemann intersection inequalities are established.

Citations

• Fixed and Periodic Points of the Intersection Body Operator
• On a non-homogeneous version of a problem of Firey
• Sharp Isoperimetric Inequalities for Affine Quermassintegrals
• Randomized isoperimetric inequalities
• Bounding marginal densities via affine isoperimetry
• Volume of the polar of random sets and shadow systems
• Shadow systems and volume of polar convex bodies

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