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

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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.