arXiv subjects

Martin Rapaport

Publications and source records attributed to Martin Rapaport.

5 recordsLinked to original sources

arXiv2026

Entropic analogues of Gr\"unbaum's inequality

The classical Gr\"unbaum inequality asserts that the proportion of the volume of a convex body cut off by a halfspace containing its barycenter is at least $1/e$. From its functional counterpart, for any log-concave random variable $X$, one has $\mathbb{P}(X\ge \mathbb{E}X)\ge 1/e$, with equality if and only if $X$ is exponential. Motivated by Gr\"unbaum's inequality for convex bodies and its functional generalizations, we prove analogous inequalities for entropy, with characterizations of the equality cases. We show that if $X$ is a log-concave random variable on $\mathbb{R}$, then $$ h(X)-\frac{e}{e-1}H_2(1/e) \leq h(X|X \leq \mathbb{E}X) \leq h(X), $$ where $h$ is the differential entropy, $H_2(\cdot)$ is the binary entropy function and $X|X\leq \mathbb{E}X$ stands for the distribution of $X$ conditional on $X\leq \mathbb{E}X$. We generalize the upper bound for all R\'enyi entropies and the lower bound for min-entropy. Our inequalities are sharp and we characterize all equality cases. We discuss potential generalizations in high dimensions and give counterexamples in some directions. As an intermediate step for the proof of the lower bound, we establish a new inequality that we prove using a technique known as degrees of freedom, combined with a standard KKT-type optimization lemma. Along the way, we characterize the equality case in a known comparison inequality between differential and min-entropy, which may be of independent interest.

math.PR↗

arXiv2025

Negative Moments of Steinhaus Sums

We prove a sharp upper bound on negative moments of sums of independent Steinhaus random variables (that is uniform on circles in the plane). Together with the series of earlier works: K\"onig-Kwapie\'n (2001), Baernstein II-Culverhouse (2002), and K\"onig (2014), this closes the investigation of sharp $L_p-L_2$ Khinchin-type inequalities for the Steinhaus sums. Incidentally, we fix a mistake in an earlier paper, as well as provide an application to sharp bounds on R\'enyi entropy.

math.PR↗

arXiv2025

Entropic versions of Bergstr\"om's and Bonnesen's inequalities

We establish analogues of the Bergstr\"om and Bonnesen inequalities, related to determinants and volumes respectively, for the entropy power and for the Fisher information. The obtained inequalities strengthen the well-known convolution inequality for the Fisher information as well as the entropy power inequality in dimensions $d>1$, while they reduce to the former in $d=1$. Our results recover the original Bergstr\"om inequality and generalize a proof of Bergstr\"om's inequality given by Dembo, Cover and Thomas. We characterize the equality case in our entropic Bonnesen inequality.

cs.IT↗

arXiv2024

On the monotonicity of discrete entropy for log-concave random vectors on $\mathbb{Z}^d$

We prove the following type of discrete entropy monotonicity for sums of isotropic, log-concave, independent and identically distributed random vectors $X_1,\dots,X_{n+1}$ on $\mathbb{Z}^d$: $$ H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n}) + \frac{d}{2}\log{\Bigl(\frac{n+1}{n}\Bigr)} +o(1), $$ where $o(1)$ vanishes as $H(X_1) \to \infty$. Moreover, for the $o(1)$-term, we obtain a rate of convergence $ O\Bigl({H(X_1)}{e^{-\frac{1}{d}H(X_1)}}\Bigr)$, where the implied constants depend on $d$ and $n$. This generalizes to $\mathbb{Z}^d$ the one-dimensional result of the second named author (2023). As in dimension one, our strategy is to establish that the discrete entropy $H(X_1+\cdots+X_{n})$ is close to the differential (continuous) entropy $h(X_1+U_1+\cdots+X_{n}+U_{n})$, where $U_1,\dots, U_n$ are independent and identically distributed uniform random vectors on $[0,1]^d$ and to apply the theorem of Artstein, Ball, Barthe and Naor (2004) on the monotonicity of differential entropy. In fact, we show this result under more general assumptions than log-concavity, which are preserved up to constants under convolution. In order to show that log-concave distributions satisfy our assumptions in dimension $d\ge2$, more involved tools from convex geometry are needed because a suitable position is required. We show that, for a log-concave function on $\mathbb{R}^d$ in isotropic position, its integral, barycenter and covariance matrix are close to their discrete counterparts. Moreover, in the log-concave case, we weaken the isotropicity assumption to what we call almost isotropicity. One of our technical tools is a discrete analogue to the upper bound on the isotropic constant of a log-concave function, which extends to dimensions $d\ge1$ a result of Bobkov, Marsiglietti and Melbourne (2022).

math.PR↗

arXiv2023

Criteria for entropic curvature on graph spaces

In this paper we establish new simple local geometric criteria for discrete entropic curvature introduced in [47] that are powerful enough to capture many geometric properties of complex models arising in mathematical physics. These results are robust in the sense that they apply to any discrete graph equipped with a Markov reversible generator. Our definitions of entropic curvature differ from the one of the pioneering works of Erbar-Maas [19,20] (which is already a discrete analog of the Lott-Sturm-Villani entropic curvature in the continuous setting). Singularly, our results provide refined concentration properties related to the celebrated convex-hull method by Talagrand [50,51] for a large class of probability measures that cannot be captured from Erbar- Maas entropic definition of curvature. Our approach gives also a new insight of the convex hull method, without being related to induction arguments. We illustrate the power of our results, as well as the general entropic strategy developed in this paper, to tackle challenging models studied in mathematical physics, including Gibbs measures with interaction potentials such as Ising models on the discrete hypercube and measures with interaction potential on the lattice $\mathbb{Z}^n$. For instance, we significantly improve the constant of the refined convex concentration properties obtained in [2] for Ising models. Moreover, when dealing with the antiferromagnetic Curie-Weiss model, we improve the previously known bound for entropic curvature by a factor of $\sqrt{n}$. Our simple criteria also provides the expected right order of magnitude $C/\sqrt n$ for the lower-bound on the entropic curvature for the renowned Sherrington-Kirkpatrick model from the spin glass theory. This last result is consistant with the recent works [6,17] on the modified logarithmic Sobolev and Poincar\'e inequalities for the Sherrington-Kirkpatrick model.

math.PR↗