arXiv research

This paper proves a curvature entropy inequality for non-symmetric convex bodies.

The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.

Modified Bakry-Émery criterion inequality for Tsallis entropy monotonicity.

Inequalities linking entropy, Fisher info, Stein discrepancy, and Wasserstein distance on Riemannian manifolds.

Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.

The abstract applies waist inequality to dynamical systems and entropy.

The paper extends entropy formulas to super Ricci flows on metric measure spaces.

New method uses entropy dissipation to prove isoperimetric inequalities.

The paper characterizes curvature-dimension conditions and related inequalities on Riemannian manifolds.

In this survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates along the Ricci flow, including the Ricci flow with surgeries.

Optimal geometric estimates for Kähler manifolds with bounded Nash entropy

Ancient Ricci flows with bounded Nash entropy have uniform Sobolev inequalities.

Paper proves generalized Talagrand inequality for Sinkhorn distance.

Sharp inequality in spaces with non-negative Ricci curvature.

Study spherical convex bodies using $L_p$-floating areas and curvature entropy.

The relative $α$-entropy is the Rényi analog of relative entropy and arises prominently in information-theoretic problems. Recent information geometric investigations on this quantity have enabled the generalization of the Cramér-Rao inequality, which provides a lower bound for the variance of an estimator of an escort…

In this paper, under the generalized curvature-dimension inequality recently introduced by F. Baudoin and N. Garofalo, we obtain differential Harnack inequalities for the positive solutions to the Schödinger equation associated to subelliptic operator with potential. As applications of the differential Harnack inequali…

In the present paper, we prove that a lower bound on the $1$-weighted Ricci curvature is equivalent to a convexity of entropies on the Wasserstein space. Based on such characterization, we provide some interpolation inequalities such as the Pr'ekopa-Leindler inequality, the Borel-Branscamp-Lieb inequality, and the Brun…

In this paper, we prove the concavity of $p$-entropy power of probability densities solving the $p$-heat equation on closed Riemannian manifold with nonnegative Ricci curvature. As applications, we give new proofs of $L^p$-Euclidean Nash inequality and $L^p$-Euclidean Logarithmic Sobolev inequality, moreover, an improv…

New proof of Wulff-Gage inequality with applications.

The paper develops inequalities for log-concave functions and related surface areas.

We generalize an entropy calculation of Perelman to the case of domains evolving inside a Ricciflow solution. In the case of Euclidean space as ambient manifold an interesting relation with Harnack inequalities emerges.

Uniform Sobolev inequality for Kähler metrics with entropy bound.

In this communication, we describe some interrelations between generalized $q$-entropies and a generalized version of Fisher information. In information theory, the de Bruijn identity links the Fisher information and the derivative of the entropy. We show that this identity can be extended to generalized versions of en…

Characterizes a new curvature bound with convexity of entropies.

Software estimates inequality in random systems with changing communities.

The paper explores inequalities for strongly-convex sets in weighted Riemannian manifolds.

We prove the concavity of $p$-Rényi entropy power for positive solutions to the doubly nonlinear diffusion equations on $\mathbb{R}^n$ or compact Riemannian manifolds with nonnegative Ricci curvature. As applications, we give new proofs of the sharp $L^p$-Sobolev inequality and $L^p$-Gagliardo-Nirenberg inequalities on…

McDiarmid's inequality under dependence via approximate tensorization of entropy

Study on finite entropy and energy in Kähler geometry.

Proposes a new criterion for selecting Nash equilibria considering both utility and inequality.

Improved bounds linking entropy and volume in hyperbolic 3-manifolds.

We derive weighted log-Sobolev inequalities from a class of super Poincaré inequalities. As an application, the Talagrand inequality with larger distances are obtained. In particular, on a complete connected Riemannian manifold, we prove that the $\log^\dd$-Sobolev inequality with $\dd\in (1,2)$ implies the $L^{2/(2-\d…

The paper proves the monotonicity of a modified Perelman's W-entropy for mean curvature flow.

The study proves surfaces with high genus have a specific inequality.

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

In this paper, we prove the characterization of the $(K, \infty)$-super Perelman Ricci flows by various functional inequalities and gradient estimate for the heat semigroup generated by the Witten Laplacian on manifolds equipped with time dependent metrics and potentials. As a byproduct, we derive the Hamilton type dim…

The paper extends entropy concepts to Monge-Ampère measures with prescribed singularities.

In this paper, we prove the concavity of the Renyi entropy power for nonlinear diffusion equation (NLDE) associated with the Laplacian and the Witten Laplacian on compact Riemannian manifolds with non-negative Ricci curvature or $CD(0,m)$-condition and on compact manifolds equipped with time dependent metrics and poten…

We establish isosystolic inequalities for a class of manifolds which includes the aspherical manifolds. In particular, we relate the systolic volume of aspherical manifolds first to their minimal entropy, then to the algebraic entropy of their fundamental groups.

Analytical results bound the approach to oligarchy in a modified asset exchange model.

The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{po…

Local Sobolev inequality on Ricci flows with applications.

In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(-K, m)$-condition, where $m\in [n, \infty)$ and $K\geq 0$ are two constants. Moreover, we introduce the $W$-entropy and prove the $W$-ent…

The purpose of this work is to study some monotone functionals of the heat kernel on a complete Riemannian manifold with nonnegative Ricci curvature. In particular, we show that on these manifolds, the gradient estimate of Li and Yau, the gradient estimate of Ni, the monotonicity of the Perelman's entropy and the volum…

New inequality links surface orthospectrum to boundary length.

Abstract: Generalizes multisymplectic forms to vector-valued versions.

We prove local Poincaré inequalities under various curvature-dimension conditions which are stable under the measured Gromov-Hausdorff convergence. The first class of spaces we consider is that of weak CD(K,N) spaces as defined by Lott and Villani. The second class of spaces we study consists of spaces where we have a …