# Intrinsic entropies of log-concave distributions

## TL;DR

This paper introduces the concept of intrinsic entropies for log-concave distributions, revealing how intrinsic volumes of typical sets grow exponentially and defining a continuous family of entropy measures.

## Contribution

It defines the intrinsic entropy function for log-concave distributions and proves its continuity, extending the classical notion of entropy through intrinsic volumes.

## Key findings

- Intrinsic entropy $h_X( heta)$ grows exponentially with dimension.
- $h_X( heta)$ is continuous over [0,1].
- Interpolates between 0 and the classical entropy $h(X)$.

## 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.