Uniform Distributions on p-Balls and the Singular Role of $p=1,2,\infty$ in p-Norm Geometry

This paper studies the relationship between volume and surface uniform measures on n-dimensional p-balls under the p-norm. It is proved that for p=1, p=2 and p=infinity, and only for these values of p, radial projection maps a volumetrically uniform distribution to a surface-uniform distribution. Algorithms for uniform sampling on p-balls and p-spheres are provided, together with empirical illustrations.