On the Geometry of Convex Typical Sets

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On the Geometry of Convex Typical Sets

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Creation Date: Jun 14, 2015

Published In: Jun 2015

Paper Type: Conference Paper

Book Title: Proceedings of the 2015 IEEE International Symposium on Information Theory

Address: Hong Kong, China

Abstract:

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 growth rate of the volume of typical sets, the exponential growth rate of intrinsic volumes generalizes the differential entropy of log-concave distributions. We conjecture a version of the entropy power inequality for such a generalization of differential entropy. 

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