The Makai inequality in higher dimensions: qualitative and quantitative aspects

TL;DR

This paper generalizes Makai's inequality to higher dimensions, relating Laplacian torsional rigidity with perimeter and measure, and provides quantitative estimates on geometric structure.

Contribution

It extends Makai's inequality from the plane to arbitrary dimensions and introduces quantitative bounds on geometric properties.

Findings

Established a sharp inequality involving torsional rigidity, perimeter, and measure in higher dimensions.

Proved the inequality generalizes Makai's original planar result.

Provided quantitative estimates revealing geometric structure and thickness of optimal sequences.

Abstract

In this paper, given a convex, bounded, open set $\Omega \subset \mathbb{R}^n$ we prove a sharp inequality involving the Laplacian torsional rigidity and both the perimeter and the measure of the domain. Our result generalizes to arbitrary dimensions the inequality established by Makai in the plane which, as conjectured in arXiv:2007.02549. Furthermore, we establish quantitative estimates that provide key insights into the geometric structure and the thickness of the underlying optimizing sequences.