Georgia Institute of Technology College of Sciences

We consider the following problem. Does there exist an absolute constant C such that for every natural number n, every integer 1 \leq k \leq n, every origin-symmetric convex body L in R^n, and every measure \mu with non-negative even continuous density in R^n, \mu(L) \leq C^k \max_{H \in Gr_{n-k}} \mu(L \cap H}/|L|^{k/n}, where Gr_{n-k} is the Grassmannian 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 hyperplace conjecture of Bourgain, a major open problem in convex geometry. We show that the above inequality holds for arbitrary origin-symmetric convex bodies, all k and all \mu with C \sim \sqrt{n}, and with an absolute constant C for some special class of bodies, including unconditional bodies, unit balls of subspaces of L_p, and others. We also prove that for every \lambda \in (0,1) there exists a constant C = C(\lambda) so that the above inequality holds for every natural number, every origin-symmetric convex body L in R^n, every measure \mu with continuous density and the codimension of sections k \geq \lambda n. The latter result is new even in the case of volume. The proofs are based on a stability result for generalized intersections bodies and on estimates of the outer volume ratio distance from an arbitrary convex body to the classes of generalized intersection bodies.