Volumes of high-dimensional balls and concentration near the boundary

” 20-dimensional watermelon”

1. Find the surface area of the unit n-dimensional sphere \(S ^ { n }\) Indication: Use the notion of Gamma function \(\Gamma ( a ) \equiv \int _ { 0 } ^ { \infty } e ^ { - x } x ^ { a - 1 } d x\)

2. Find the volume \(V _ { n }\) of the unit n-dimensional ball \(B ^ { n }\)

3. Denote by \(V _ { n , a }\) the volume of a layer of depth a adjacent to the surface of \(B ^ { n } , 0 < a < 1\) . Compute the ratio \(\frac { V _ { n , a } } { V _ { n } }\) and investigate the limit \(n \to \infty\)

4. Assuming the watermelon rind has a thickness 0.1 of its radius, find what fraction of the total volume of the 20-dimensional watermelon would be occupied by its rind.