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

2024/11/11 by Pinzón, Carlos
#Cryptography and Security (cs.CR) #FOS: Computer and information sciences #FOS: Mathematics #Numerical Analysis (math.NA) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2411.13567

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.

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