Interesting Properties of Hyperspheres

One property of high-dimensional $d$ spaces (such as $p=(\vec{p}_1, \vec{p}_2,\ldots, \vec{p}_N)$-space of dimension $d=3N$ occurring in phase space integrals encountered in the statistical mechanics of $N$ particles) is that “the volume of a $d$-dimensional body is concentrated near its surface.” In my class on graduate statistical thermodynamics, we explored this phenomenon in the following problem.

Consider a $d$-dimensional ball of radius $1$. Calculate the fraction of the volume concentrated between radius $1$ and $1-\varepsilon$. How does this behave when $d$ becomes very large while $\varepsilon$ is kept fixed (even if small)? How does this behave when $d$ becomes very large while at the same time adjusting $\varepsilon=x/d$ with fixed $x$?

Consider a $50$-dimensional watermelon with a radius of 20 cm and skin that’s 1 cm thick. What is the fraction of the volume contained in the skin?

This page is currently under construction.