Paper proves generalized Talagrand inequality for Sinkhorn distance.
problem Proving a generalized Talagrand inequality for Sinkhorn distance.
method Using entropy power inequality and infinitesimal displacement convexity of optimal transport map.
result Extends previous results of Gaussian Talagrand inequality for Sinkhorn distance to strongly log-concave case.
The paper extends entropy formulas to super Ricci flows on metric measure spaces.
problem Entropy formulas for super Ricci flows on metric measure spaces.
method Extending Perelman's W-entropy and Shannon entropy power to super Ricci flows. result Equivalence between volume non-local collapsing property and lower boundedness of W-entropy on RCD(0,N) spaces. 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 Lp-Euclidean Nash inequality and Lp-Euclidean Logarithmic Sobolev inequality, moreover, an improv…
We prove the concavity of p-Rényi entropy power for positive solutions to the doubly nonlinear diffusion equations on Rn or compact Riemannian manifolds with nonnegative Ricci curvature. As applications, we give new proofs of the sharp Lp-Sobolev inequality and Lp-Gagliardo-Nirenberg inequalities on…
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…
The paper characterizes curvature-dimension conditions and related inequalities on Riemannian manifolds.
problem Curvature-dimension conditions and related inequalities on Riemannian manifolds.
method Information-theoretic approach to study curvature-dimension condition, rigidity theorems, and entropy differential inequalities.
result Equivalence of curvature-dimension condition and entropy differential inequalities on Riemannian manifolds.
In this paper, we prove the concavity of the Shannon entropy power for the heat equation associated with the Laplacian or the Witten Laplacian on complete Riemannian manifolds with suitable curvature-dimension condition and on compact super Ricci flows. Under suitable curvature-dimension condition, we prove that the ri…
The study establishes a curvature-dimension condition for discrete Markov chains.
problem Proving modified logarithmic Sobolev inequalities for discrete Markov chains.
method Identifying and proving a curvature-dimension inequality CDΥ(κ,∞), and showing its compatibility with diffusive settings. result The CDΥ condition preserves curvature bounds under tensorization and leads to Beckner inequalities. This paper proves a curvature entropy inequality for non-symmetric convex bodies.
problem Proving a curvature entropy inequality for non-symmetric convex bodies.
method Demonstrated the log-Minkowski inequality of curvature entropy for general convex bodies in 2D.
result Equivalence of cone-volume measure uniqueness, log-Minkowski volume inequality, and curvature entropy inequality for general convex bodies in 2D.
The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.
problem Proving Li-Yau inequalities and modified logarithmic Sobolev inequalities for reversible Markov chains.
method Introducing the CDΥ(κ,F) condition and deriving entropy-information inequalities. result Derives functional inequalities relating entropy to Fisher information.
Modified Bakry-Émery criterion inequality for Tsallis entropy monotonicity.
problem Establishing improved logarithmic Sobolev inequalities and monotonicity of Tsallis entropy.
method Proving a one-parameter family of weighted Bakry-Émery Γ2 criterion inequalities and a modified inequality. result Yields a family of sharp Sobolev inequalities and monotonicity of Tsallis entropy.
Inequalities linking entropy, Fisher info, Stein discrepancy, and Wasserstein distance on Riemannian manifolds.
problem Linking entropy, Fisher info, Stein discrepancy, and Wasserstein distance on Riemannian manifolds.
method Deriving inequalities linking these measures on Riemannian manifolds.
result Strengthening and extending existing inequalities to Riemannian manifolds.
Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.
problem Proving an isoperimetric inequality on Finsler metric measure manifolds.
method Defining volume entropy and second Cheeger constant, proving sharp inequality.
result Sharp isoperimetric inequality involving volume entropy and weighted Ricci curvature.
The abstract applies waist inequality to dynamical systems and entropy.
problem Understanding the relationship between waist inequality and dynamical systems.
method Applying waist inequality to entropy and mean dimension of dynamical systems.
result Maps between dynamical systems have positive conditional metric mean dimension under certain conditions.
New method uses entropy dissipation to prove isoperimetric inequalities.
problem Proving isoperimetric inequalities in geometric settings.
method Information-theoretic approach based on entropy dissipation under heat flow.
result New proof of Euclidean isoperimetric inequality with sharp constant.
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
problem Optimal geometric estimates for compact Kähler manifolds
method Proving Sobolev-type inequality and local volume noncollapsing with optimal exponents
result Uniformly bounded q-Nash entropy Ancient Ricci flows with bounded Nash entropy have uniform Sobolev inequalities.
problem Bounding Nash entropy in ancient Ricci flows.
method Uniformly bounded Nash entropy implies uniform bounds on the ν-functional, leading to uniform logarithmic and Sobolev inequalities.
result Uniform logarithmic and Sobolev inequalities on ancient Ricci flows with bounded Nash entropy.
Behavior of the entropy numbers of classes of multivariate functions with mixed smoothness is studied here. This problem has a long history and some fundamental problems in the area are still open. The main goal of this paper is to develop a new method of proving the upper bounds for the entropy numbers. This method is…
Sharp inequality in spaces with non-negative Ricci curvature.
problem Proving a sharp isoperimetric inequality in metric measure spaces.
method Using volume entropy in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
result Proved a sharp dimension-free isoperimetric inequality.
Study spherical convex bodies using Lp-floating areas and curvature entropy.
problem Analogous isoperimetric inequalities for spherical convex bodies.
method Introduced Lp-floating areas and curvature entropy for spherical convex bodies. result Established isoperimetric inequalities and dual isoperimetric inequalities.
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…
New proof of Wulff-Gage inequality with applications.
problem Proving the Wulff-Gage isoperimetric inequality.
method Provided a new proof of the inequality for origin-symmetric convex bodies.
result Uniqueness of log-Minkowski problem and new proof of log-Minkowski inequality.
The paper develops inequalities for log-concave functions and related surface areas.
problem Understanding log-concave functions and their inequalities.
method Establishing new inequalities through f-divergences and functional affine surface areas.
result New inequalities on functional affine surface area and bounds for Kullback-Leibler divergence.
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.
problem Establishing Sobolev inequalities for Kähler metrics with entropy bound.
method Uniform Sobolev inequality for Kähler metrics with entropy bound and no lower Ricci curvature bound.
result Derive various geometric estimates for Kähler-Einstein currents.
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.
problem Lowering Ricci curvature bounds with ε-range.
method Characterization through convexity of entropies over Wasserstein space.
result Derives various interpolation and functional inequalities.
Software estimates inequality in random systems with changing communities.
problem Measuring inequality in systems with dynamic interactions and random attributes.
method Piecewise homogeneous Markov chain for changing points, copula function for multivariate distribution, Monte Carlo algorithm for entropy estimation.
result Estimates Random Theil's Entropy to measure inequality in random systems.
The paper explores inequalities for strongly-convex sets in weighted Riemannian manifolds.
problem Investigating dilation type inequalities on weighted Riemannian manifolds.
method Introducing dilation profile and comparing it with model space under lower weighted Ricci curvature bounds.
result Showed several functional inequalities related to various entropies.
McDiarmid's inequality under dependence via approximate tensorization of entropy
problem Dependent versions of McDiarmid's inequality
method Approximate tensorization of entropy (ATE)
result Derives McDiarmid's inequality for non-isotropic Gaussian random vectors
Study on finite entropy and energy in Kähler geometry.
problem Finite entropy and energy measures in Kähler geometry.
method Refined Moser-Trudinger inequalities for quasi-plurisubharmonic functions.
result Quasi-plurisubharmonic potentials with finite entropy belong to the finite energy class En−1n. Proposes a new criterion for selecting Nash equilibria considering both utility and inequality.
problem Finding a fair Nash equilibrium in group decision-making.
method Introduces entropy-norm space for geometric selection of strict Nash equilibria.
result The closest entropy-norm pair to the largest entropy-norm pair in rescaled space is the most suitable equilibrium.
We study the geometry of probability distributions with respect to a generalized family of Csiszár f-divergences. A member of this family is the relative α-entropy which is also a Rényi analog of relative entropy in information theory and known as logarithmic or projective power divergence in statistics. We apply E…
Improved bounds linking entropy and volume in hyperbolic 3-manifolds.
problem Establishing bounds between entropy and volume in hyperbolic 3-manifolds.
method Heegaard Floer homology and hyperbolic geometry.
result Entropy is bounded by hyperbolic volume with logarithmic factor.
Ancient flows by curvature powers in 2D have finite entropy.
problem Existence of non-homothetic ancient flows by powers of curvature in R2. method Determined Morse indices and kernels of the linearized operator of shrinkers. Constructed flows using unstable eigenfunctions.
result Existence of ancient flows with finite entropy.
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.
problem Proving the monotonicity of Perelman's W-entropy for mean curvature flow.
method Modified definition of K. Ecker's W-entropy and Hamilton's Harnack inequality.
result The modified W-entropy is monotonically decreasing in time.
The study proves surfaces with high genus have a specific inequality.
problem Proving surfaces with high genus satisfy a specific inequality.
method Using volume entropy and systolic ratio inequality.
result Every closed surface of genus at least 18 satisfies Loewner's systolic ratio inequality.
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing Lp relative surface areas, proving invariance and inequalities, and using geometric interpretations. result Established inequalities and a new notion of entropy for ball-convex bodies.
In this paper, we prove the characterization of the (K,∞)-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.
problem Investigating entropy for Monge-Ampère measures with specific singularities.
method Generalizing entropy for potentials, studying stability under blow-ups and perturbations, proving Moser-Trudinger inequalities.
result Functions with finite entropy belong to a specific energy class and maintain singularities of the model potential.
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.
problem Analyzing economic inequality in a modified asset exchange model.
method Analytical results using Gini coefficient and differential inequalities.
result The Gini coefficient is bounded by a first-order differential inequality.
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.
problem Understanding the local geometry of Ricci flows.
method Proving a local Sobolev inequality for Ricci flows.
result The local ν-functional depends only on the Nash entropy at the center of a disk.