The Busemann-Petty problem for arbitrary measures

摘要

The Busemann-Petty problem asks whether symmetric convex bodies in ℝ n with smaller (n−1)-dimensional volume of central hyperplane sections necessarily have smaller n-dimensional volume. The answer to this problem is affirmative for n≤4 and negative for n≥5. In this paper we generalize the Busemann-Petty problem to essentially arbitrary measure in place of the volume. We also present applications of the latter result by proving several inequalities concerning the measure of sections of convex symmetric bodies in ℝ n .