A [formula omitted] estimate for measures of hyperplane sections of convex bodies

Abstract

The hyperplane (or slicing) problem asks whether there exists an absolute constant C so that for any origin-symmetric convex body K in Rn|K|n−1n⩽Cmaxξ∈Sn−1|K∩ξ⊥|, where ξ⊥ is the central hyperplane in Rn perpendicular to ξ, and |K| stands for volume of proper dimension. The problem is still open, with the best-to-date estimate C∼n1/4 established by Klartag, who slightly improved the previous estimate of Bourgain. It is much easier to get a weaker estimate with C=n.In this note we show that the n estimate holds for arbitrary measure in place of volume. Namely, if L is an origin-symmetric convex body in Rn and μ is a measure with non-negative even continuous density on L, thenμ(L)⩽nnn−1cnmaxξ∈Sn−1μ(L∩ξ⊥)|L|1/n, where cn=|B2n|n−1n/|B2n−1|<1, and B2n is the unit Euclidean ball in Rn. We deduce this inequality from a stability result for intersection bodies.