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

TL;DR

This paper explores the unique geometric properties of p-balls in n-dimensional space, highlighting the special roles of p=1, 2, and infinity in the relationship between volume and surface measures, and provides algorithms for uniform sampling.

Contribution

It establishes the singular role of p=1, 2, and infinity in the geometry of p-balls and introduces algorithms for uniform sampling on these shapes.

Findings

Radial projection maps volume to surface uniformity only for p=1, 2, and infinity.

Algorithms for uniform sampling on p-balls and p-spheres are developed.

Empirical illustrations demonstrate the sampling algorithms' effectiveness.

Abstract

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.