论文标题
高维欧几里得球的体积衰减和浓度 - PDE和变化透视
Volume decay and concentration of high-dimensional Euclidean balls -- a PDE and variational perspective
论文作者
论文摘要
这是一个众所周知的事实 - 可以通过基础演算表明 - $ \ mathbb {r}^n $衰减中的单位球的体积降低到零，并同时将其集中在边界球附近的薄外壳上，为$ n \ nearrow \ nearrow \ indrow \ infty $。从不同的观点，包括欧几里得几何形状，凸几何，Banach空间理论，组合，概率，离散的几何形状等，为这一事实提供了许多严格的证据和启发式论证。在此注释中，我们通过椭圆形偏微分方程和变量计算的规律性理论提供了两个证据。
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.
