Sharp moment-entropy inequalities and capacity bounds for log-concave distributions

Abstract

We show that the uniform distribution minimizes entropy among all one-dimensional symmetric log-concave distributions with fixed variance, as well as various generalizations of this fact to Rényi entropies of orders less than 1 and with moment constraints involving $p$-th absolute moments with $p\le 2$. As consequences, we give new capacity bounds for additive noise channels with symmetric log-concave noises, as well as for timing channels involving positive signal and noise where the noise has a decreasing log-concave density. In particular, we show that the capacity of an additive noise channel with symmetric, log-concave noise under an average power constraint is at most 0.254 bits per channel use greater than the capacity of an additive Gaussian noise channel with the same noise power. Consequences for reverse entropy power inequalities and connections to the slicing problem in convex geometry are also discussed.

arXiv commentthe original paper split into two: this one and arXiv:1904.02314; additional results added to this part

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Cited by in corpus (3)

• Bernoulli sums and Rényi entropy inequalities
• Entropy-variance inequalities for discrete log-concave random variables via degree of freedom
• Entropies of sums of independent gamma random variables