High Dimensional Space Oddity

Music City Center 

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).