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.

翻译：本文研究n维p-球在p-范数下体积均匀测度与表面均匀测度之间的关系。研究证明，当且仅当p=1、p=2和p=∞时，径向投影映射能将体积均匀分布转换为表面均匀分布。本文提供了p-球和p-球面上均匀采样的算法，并辅以实证示例说明。