高维欧几里得球体的体积衰减与集中——一个PDE和变分视角
Volume decay and concentration of high-dimensional Euclidean balls -- a PDE and variational perspective
AI总结：
本文从PDE和变分法视角，给出了高维欧几里得球体体积衰减与集中现象的两个新证明。
摘要：
这是一个众所周知的事实——可以通过初等微积分证明——单位球在$\mathbb{R}^n$中的体积随着$n \nearrow \infty$趋于零，同时集中于近边界球的薄壳上。许多严谨的证明和启发式论证从不同视角提供了这一事实，包括欧几里得几何、凸几何、Banach空间理论、组合学、概率、离散几何等。在本文中，我们通过椭圆偏微分方程的正则性理论和变分法给出了另一种两个证明。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
