Volume decay and concentration of high-dimensional Euclidean balls -- a PDE and variational perspective

TL;DR

This paper presents two novel proofs, using PDE regularity theory and calculus of variations, for the high-dimensional phenomenon where the volume of Euclidean balls concentrates near the boundary as dimension increases.

Contribution

It introduces new proof techniques for the volume decay and concentration phenomenon in high-dimensional Euclidean balls, connecting geometric facts with PDE and variational methods.

Findings

Volume of high-dimensional Euclidean balls concentrates near the boundary.

New proofs via elliptic PDE regularity theory and calculus of variations.

Provides alternative perspectives on a classical geometric fact.

Abstract

It is a well-known fact -- which can be shown by elementary calculus -- that the volume of the unit ball in $\mathbb{R}^n$ decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as $n \nearrow \infty$. Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.