This paper investigates the rate-distortion function, under a squared error distortion $D$, for an $n$-dimensional random vector uniformly distributed on an $(n-1)$-sphere of radius $R$. First, an expression for the rate-distortion function is derived for any values of $n$, $D$, and $R$. Second, two types of asymptotics with respect to the rate-distortion function of a Gaussian source are characterized. More specifically, these asymptotics concern the low-distortion regime (that is, $D \to 0$) and the high-dimensional regime (that is, $n \to \infty$).

翻译：本文研究半径为$R$的$(n-1)$-球面上均匀分布的$n$维随机向量在平方误差失真$D$下的率失真函数。首先，针对任意$n$、$D$和$R$值，推导出率失真函数的表达式。其次，刻画了与高斯信源率失真函数相关的两种渐近特性。具体而言，这些渐近特性涉及低失真情形（即$D \to 0$）和高维情形（即$n \to \infty$）。