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

摘要

The Busemann-Petty problem for an arbitrary measure with non-negative even continuous density in asks whether origin-symmetric convex bodies in with smaller -dimensional measure of all central hyperplane sections necessarily have smaller measure It was shown in Zvavitch (Math Ann 331:867-887, 2005) that the answer to this problem is affirmative for and negative for . In this paper we prove an isomorphic version of this result. Namely, if are origin-symmetric convex bodies in such that for every then Here is the central hyperplane perpendicular to We also study the above question with additional assumptions on the body and present the complex version of the problem. In the special case where the measure is convex we show that can be replaced by where is the maximal isotropic constant. Note that, by a recent result of Klartag, . Finally we prove a slicing inequality for any convex even measure and any symmetric convex body in where is an absolute constant. This inequality was recently proved in Koldobsky (Adv Math 254:33-40, 2014) for arbitrary measures with continuous density, but with in place of n(1/4).