Slicing inequalities for measures of convex bodies

Abstract

We consider the following problem. Does there exist an absolute constant C so that for every n∈N, every integer 1≤k<n, every origin-symmetric convex body L in Rn, and every measure μ with non-negative even continuous density in Rn,(1)μ(L)≤CkmaxH∈Grn−k⁡μ(L∩H)|L|k/n, where Grn−k is the Grassmanian of (n−k)-dimensional subspaces of Rn, and |L| stands for volume? This question is an extension to arbitrary measures (in place of volume) and to sections of arbitrary codimension k of the slicing problem, a major open problem in convex geometry.It was proved in [18,19] that (1) holds for arbitrary origin-symmetric convex bodies, all k and all μ with C≤O(n). In this article, we prove inequality (1) with an absolute constant C for unconditional convex bodies and for duals of bodies with bounded volume ratio. We also prove that for every λ∈(0,1) there exists a constant C=C(λ) so that inequality (1) holds for every n∈N, every origin-symmetric convex body L in Rn, every measure μ with continuous density and the codimension of sections k≥λn. The proofs are based on a stability result for generalized intersection bodies and on estimates of the outer volume ratio distance from an arbitrary convex body to the classes of generalized intersection bodies. In the last section, we show that for some measures the behavior of minimal sections may be very different from the case of volume.