May 21, 2014 AN ISOMORPHIC VERSION OF THE BUSEMANN-PETTY PROBLEM FOR ARBITRARY MEASURES

The Busemann-Petty problem for an arbitrary measure μ with non-negative even continuous density in R asks whether origin-symmetric convex bodies in R with smaller (n − 1)-dimensional measure μ of all central hyperplane sections necessarily have smaller measure μ. It was shown in [Zv] that the answer to this problem is affirmative for n ≤ 4 and negative for n ≥ 5. In this paper we prove an isomorphic version of this result. Namely, if K,M are origin-symmetric convex bodies in R such that μ(K∩ξ⊥) ≤ μ(M ∩ξ⊥) for every ξ ∈ Sn−1, then μ(K) ≤ √ n μ(M). Here ξ⊥ is the central hyperplane perpendicular to ξ. We also study the above question with additional assumptions on the body K and present the complex version of the problem. In the special case where the measure μ is convex we show that √ n can be replaced by cLn, where Ln is the maximal isotropic constant. Note that, by a recent result of Klartag, Ln ≤ O(n). Finally we prove a slicing inequality μ(K) ≤ Cn max ξ∈Sn−1 μ(K ∩ ξ) voln(K) 1 n for any convex even measure μ and any symmetric convex body K in R, where C is an absolute constant. This inequality was recently proved in [K2] for arbitrary measures with continuous density, but with √ n in place of n.