Further unifying two approaches to the hyperplane conjecture

TL;DR

This paper unifies two recent approaches to bounding the isotropic constants of convex bodies by introducing a new hereditary parameter, extending the applicability of the more powerful method across a broader range of assumptions.

Contribution

It introduces a new hereditary parameter for isotropic measures, unifying and extending previous methods to provide bounds on isotropic constants.

Findings

The new parameter relates to marginals with bounded isotropic constants.

The method in [KM] can be extended under weaker assumptions.

Unified approach broadens the scope of bounds for isotropic constants.

Abstract

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