# 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.