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An isomorphic version of the Busemann-Petty problem for arbitrary measures

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arxiv 1405.0567 v2 pith:5EUBDIKY submitted 2014-05-03 math.FA math.MG

Alexander Koldobsky , Artem Zvavitch This is my paper · ORCID
classification math.FAmath.MG
keywords measuresprovebodiesconvexhyperplaneversionarbitrarybetter
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abstract

We prove the following theorem. Let $\mu$ be a measure on $R^n$ with even continuous density, and let $K,L$ be origin-symmetric convex bodies in $R^n$ so that $\mu(K\cap H)\le \mu(L\cap H)$ for any central hyperplane H. Then $\mu(K)\le \sqrt{n} \mu(L).$ We also prove this result with better constants for some special classes of measures and bodies. Finally, we prove a version of the hyperplane inequality for convex measures.

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

    math.FA 2026-08 conditional novelty 7.0 of 10

    The optimal constant in the isomorphic Busemann-Petty problem for arbitrary even densities has the sharp order √n: a new lower bound C_n ≥ c√n matches the known upper bound C_n ≤ √n.

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