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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 $h_{X}(\theta )$ ...
We consider convex sets obtained as one-sided typical sets of log-concave distributions, and show that the sequence of logarithms of intrinsic volumes corresponding to these typical sets converges to a limit function under an appropriate scaling. The limit function may be used to represent the exponential growth rate of intrinsic volumes of the typical sets. Since differential entropy is the exponential...
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