Slicing inequalities for measures of convex bodies

Abstract

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 R n , and every measure µ with non-negative even continuous density in R n ,where Gr n−k is the Grassmanian of (n − k)-dimensional subspaces of R n , 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 of Bourgain, a major open problem in convex geometry.It was proved in [K4, K5] that (1) holds for arbitrary originsymmetric 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 R n , 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.

Estimates for moments of general measures on convex bodies

“…In particular, if K is the unit ball of an n-dimensional subspace of L p , p > 2, then S n,K ≤ C √ p with some absolute constant C; see [15]. These results are implied by the following estimate proved in [14]:…”

“…Moreover, for several classes of centrally-symmetric convex bodies, it is known that the distance d ovr (K, L n p ) is bounded by absolute constants. These classes include duals of bodies with bounded volume ratio (see [14]) and the unit balls of normed spaces that embed in L q , 1 ≤ q < ∞ (see [18,15]). In the case p = 1, they also include all unconditional convex bodies [14].…”

“…These classes include duals of bodies with bounded volume ratio (see [14]) and the unit balls of normed spaces that embed in L q , 1 ≤ q < ∞ (see [18,15]). In the case p = 1, they also include all unconditional convex bodies [14]. The proofs in these papers estimate the distance from the class of intersection bodies, but the actual bodies used there (the Euclidean ball for p > 1 and the cross-polytope for p = 1) also belong to the classes L n p , so the same arguments work for L n p .…”

“…The slicing problem for arbitrary measures was introduced in [12] and considered in [13,14,15,5,9]. In analogy with the original problem, for a centrally-symmetric convex body K ⊆ R n , let S n,K be the smallest positive number S satisfying…”

“…However, for many classes of bodies, including intersection bodies [12] and unconditional convex bodies [14], the quantity S n,K turns out to be bounded by an absolute constant. In particular, if K is the unit ball of an n-dimensional subspace of L p , p > 2, then S n,K ≤ C √ p with some absolute constant C; see [15].…”