Comments on the floating body and the hyperplane conjecture

TL;DR

This paper reformulates the hyperplane conjecture using floating bodies and establishes bounds on the distance between convex bodies and their floating bodies, advancing understanding of the slicing problem.

Contribution

It introduces a new reformulation of the hyperplane conjecture in terms of floating bodies and provides bounds on their Hausdorff distance.

Findings

Bounds on the Hausdorff distance between convex bodies and floating bodies.

Reformulation of the hyperplane conjecture using floating bodies.

Insights into the structure of convex bodies related to the slicing problem.

Abstract

We provide a reformulation of the hyperplane conjecture (the slicing problem) in terms of the floating body and give upper and lower bounds on the logarithmic Hausdorff distance between an arbitrary convex body $K\subset \mathbb{R}^{d}$\ and the convex floating body $K_{\delta}$ inside $K$.