고차원 볼의 적도 부근의 체적 분포에 대한 정밀 근사

Abstract

In high-dimensional space, the volume of an n-dimensional ball defined in Euclidean space tends to concentrate near the surface and the equator of the ball. The purpose of this paper is to show that the volume distribution of a high-dimensional ball near the equator can be accurately approximated with a standard normal distribution. Based on the volume formulas of an n-dimensional ball, Stirling’s approximation and binomial approximation are used for a straightforward derivation of the accurate approximation of the ball distribution. Numerical data are given to demonstrate the non-intuitive phenomenon of the volume concentration and the accuracy of the volume distribution approximation. At the end, some discussions are made on the implications of the volume distribution of a high-dimensional ball toward the data search problem in high-dimensional space.