Further Unifying Two Approaches to the Hyperplane Conjecture

We compare and combine two approaches that have been recently introduced by Dafnis and Paouris [7] and by Klartag and Milman [13] with the aim of providing bounds for the isotropic constants of convex bodies. By defining a new hereditary parameter for all isotropic logarithmically concave measures, we are able to show that the method in [13], and the apparently stronger conclusions it leads to, can be extended in the full range of the “weaker” assumptions of Dafnis and Paouris [7]. The new parameter we define is related to the highest dimension k≤n−1 in which one can always find marginals of an n-dimensional isotropic log-concave measure that have bounded isotropic constant.