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

Carlos Pinzón

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arXiv · 2411.13567

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

Carlos Pinzón

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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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Carlos Pinzón. 2024-11-11. Uniform Distributions on p-Balls and the Singular Role of $p=1,2,\infty$ in p-Norm Geometry. https://arxiv.org/abs/2411.13567

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