Approximately gaussian marginals and the hyperplane conjecture

TL;DR

This paper explores the relationships between major open problems in high-dimensional convex geometry, demonstrating that the thin shell conjecture implies the hyperplane conjecture, thus advancing understanding of these interconnected conjectures.

Contribution

The paper establishes that the thin shell conjecture implies the hyperplane conjecture, extending previous results linking the spectral gap conjecture to the hyperplane conjecture.

Findings

Thin shell conjecture implies hyperplane conjecture

Extension of K. Ball's result connecting spectral gap and hyperplane conjectures

Provides new insights into the structure of high-dimensional convex bodies

Abstract

We discuss connections between certain well-known open problems related to the uniform measure on a high-dimensional convex body. In particular, we show that the "thin shell conjecture" implies the "hyperplane conjecture". This extends a result by K. Ball, according to which the stronger "spectral gap conjecture" implies the "hyperplane conjecture".