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

2014/05/03 by Koldobsky, Alexander, Zvavitch, Artem
#52A20 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1405.0567

Abstract

We prove the following theorem. Let μ be a measure on Rⁿ with even continuous density, and let K,L be origin-symmetric convex bodies in Rⁿ so that μ(K∩ H)≤ μ(L∩ H) for any central hyperplane H. Then μ(K)≤ √(n) μ(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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