Abstract. A remarkable conjecture of Feige [ SIAM J. Comput., 35 (2006), pp. 964–984] asserts that for any collection of [Formula: see text] independent nonnegative random variables [Formula: see text], each with expectation at most 1, [Formula: see text], where [Formula: see text]. In this paper, we investigate this conjecture for the class of discrete log-concave probability distributions, and we prove a strengthened version. More specifically, we show that the conjectured bound [Formula: see text] holds when [Formula: see text]’s are independent discrete log-concave with arbitrary expectation.

We establish a central limit theorem for Rényi entropies when the Rényi parameters belong to (0, 1) for a large class of random vectors. This complements a celebrated result of Barron (1986). As an application, we show that a general Rényi entropy power inequality fails when the Rényi parameter is in (0, 1).

In this paper, we investigate the role of convexity in entropy power inequalities. We establish Rényi entropy power inequalities of order r ∈ (0, 1) for a large class of densities, the so-called s-concave densities. This extends recent works on Rényi entropy power inequalities.

Using a sharp version of the reverse Young inequality, and a Rényi entropy comparison result due to Fradelizi, Madiman, and Wang (2016), the authors derive Rényi entropy power inequalities for log-concave random vectors when Rényi parameters belong to [0, 1]. Furthermore, the estimates are shown to be sharp up to absolute constants.

The entropy power inequality (EPI) has a fundamental role in Information Theory, and has deep connections with famous geometric inequalities. In particular, it is often compared to the Brunn-Minkowski inequality in convex geometry. In this article, we further strengthen the relationships between the EPI and geometric inequalities. Specifically, we establish an equivalence between a strong form of reverse EPI and the hyperplane conjecture, which is a long-standing conjecture in high-dimensional convex geometry. We also provide a simple proof of the hyperplane conjecture for a certain class of distributions, as a straightforward consequence of the EPI.

Using a sharp version of the reverse Young inequality, and a Renyi entropy comparison result due to Fradelizi, Madiman, and Wang, the authors derive a Renyi entropy power inequality for log-concave random vectors when Renyi parameters belong to [0, 1]. A discussion of symmetric decreasing rearrangements of random variables strengthens the inequality and guides the exploration as to its sharpness.

We derive a lower bound on the differential entropy for symmetric log-concave random variable X in terms of the p-th absolute moment of X, which shows that entropy and p-th absolute moment of a symmetric log-concave random variable are comparable. We apply our bound to study the rate distortion function under distortion measure |x - x̂|rfor sources that follow a log-concave probability distribution. In particular, we establish that the difference between the rate distortion function and the Shannon lower bound is at most log(√2e) ≈ 1.9 bits, independently of r and the target distortion d. For mean-square error distortion, the difference is at most log √πe ≈ 1.55 bits, regardless of d. Our results generalize to the case of vector X. Our proof technique leverages tools from convex geometry.

An extension of the entropy power inequality to the form Nrα(X + Y) ≥ Nrα(X) + Nrα(Y) with arbitrary independent summands X and Y in Rnis obtained for the Rényi entropy and powers α ≥ (r + 1)/2.