High Dimensional Space Oddity

TL;DR

This paper explores the geometric nature of the Law of Large Numbers in high-dimensional spaces, illustrating how measure concentration links geometry and probability through classical probabilistic tools.

Contribution

It demonstrates how standard probabilistic methods can solve complex geometric problems in high-dimensional Euclidean spaces, emphasizing the connection between geometry and measure concentration.

Findings

LLN and CLT can be applied to high-dimensional geometric problems

Measure concentration explains geometric phenomena in high dimensions

Probabilistic tools simplify complex geometric analyses

Abstract

In his 1996 paper, Talagrand highlighted that the Law of Large Numbers (LLN) for independent random variables can be viewed as a geometric property of multidimensional product spaces. This phenomenon is known as the concentration of measure. To illustrate this profound connection between geometry and probability theory, we consider a seemingly intractable geometric problem in multidimensional Euclidean space and solve it using standard probabilistic tools such as the LLN and the Central Limit Theorem (CLT).