Intrinsic entropies of log-concave distributions

Abstract

The entropy of a random variable is well-known to equal the exponential growth rate of the volumes of its typical sets. In this paper, we show that for any log-concave random variable $X$, the sequence of the $\lfloor n\theta \rfloor^{\text{th}}$ intrinsic volumes of the typical sets of $X$ in dimensions $n \geq 1$ grows exponentially with a well-defined rate. We denote this rate by $h_X(\theta)$, and call it the $\theta^{\text{th}}$ intrinsic entropy of $X$. We show that $h_X(\theta)$ is a continuous function of $\theta$ over the range $[0,1]$, thereby providing a smooth interpolation between the values 0 and $h(X)$ at the endpoints 0 and 1, respectively.