Isomorphic Busemann--Petty for arbitrary measures: the sharp order

2026/08/03 by Alexander Koldobsky, Artem Zvavitch
Mathematics · #math.FA #msc:52A20 #msc:52A40

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arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

Let C_n be the optimal constant with the following property. For every even, continuous, strictly positive density f on Rⁿ and all origin-symmetric convex bodies K,L⊂ Rⁿ, the inequalities ∫_K∩ξ^⊥f ≤ ∫_L∩ξ^⊥f \qquadfor all ξ∈ S^n-1 imply ∫_Kf≤ C_n∫_Lf. In an earlier paper the authors proved that C_n≤√ n. In this paper, we prove the matching lower bound C_n≥ c√ n. To simplify the exposition, we first give a complete one-scale construction, based on earlier work of Klartag and Koldobsky, which yields C_n≥ c√(n/log n). For the sharp result, we use the random-rounding construction of Klartag and Livshyts as a black box and combine it with a spherical-averaging support-separation argument.

Citations

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