The paper extends inequalities for projection bodies to arbitrary measures.
problem Sharp bounds for volume ratios of convex bodies and their projection bodies.
method Generalizations of Zhang's inequality to arbitrary measures and extensions of the projection body operator.
result New Zhang-type inequalities for arbitrary measures and functions.
Innovative inequalities for divergences with applications in PAC-Bayesian bounds and Monte Carlo.
problem Developing new inequalities for divergences.
method Introducing novel change of measure inequalities for f-divergences and α-divergences. result Applications in PAC-Bayesian bounds and Monte Carlo estimates.
The paper establishes inequalities and gradient estimates for harmonic functions on Finsler measure spaces.
problem Functional and geometric inequalities on Finsler measure spaces.
method Local uniform Poincaré and Sobolev inequalities, mean value inequality, Harnack inequalities, and gradient estimates.
result Global gradient estimates for positive harmonic functions on Finsler measure spaces.
Sharp isoperimetric inequality proven for specific metric measure spaces.
problem Proving isoperimetric inequality in metric measure spaces with synthetic conditions.
method Synthetic condition called Measure-Contraction property; Lévy-Gromov inequality.
result Sharp isoperimetric inequality holds true for spaces with synthetic conditions.
Unified proof of inequality for metric measure spaces with lower Ricci curvature bounds.
problem Establishing a Fenchel-Willmore-Chen inequality for metric measure spaces.
method Using a lower bound on the weighted intermediate Ricci curvature, extending previous results.
result Unified proof of the Fenchel-Willmore-Chen inequality.
The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.
problem Proving a Harnack inequality for positive solutions to heat equations on Finsler metric measure manifolds.
method Volume comparison theorem, weighted Poincaré inequality, local uniform Sobolev inequality, mean value inequalities.
result Derives a Harnack inequality for positive solutions to heat equations.
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.
The paper proves metric measure spaces with specific inequalities have n-dimensional volume growth.
problem Proving geometric and topological properties of spaces with Caffarelli-Kohn-Nirenberg inequalities.
method Volume doubling condition and Caffarelli-Kohn-Nirenberg inequality with same exponent n.
result Metric measure spaces with these inequalities have exactly n-dimensional volume growth.
On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-Émery Ricci curvature lower bound, modified Laplacian and the measure of certain special sets. We apply this estimate to prove Harnack inequalities for the modified Laplacian operator and fully non-lin…
We consider concepts and models for measuring inequality in the distribution of resources with a focus on how inequality varies as a function of covariates. Lorenz introduced a device for measuring inequality in the distribution of income that indicates how much the incomes below the uth quantile fall short of the…
Uniform rectifiability proven for sets with Poincaré inequalities.
problem Uniform rectifiability of sets with Poincaré inequalities.
method Weak (1,d)-Poincaré inequality and surface measure. result Uniform rectifiability achieved for sets supporting such inequalities.
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.
The paper establishes new inequalities for Finsler measure spaces.
problem Developing inequalities for Finsler measure spaces.
method Study of linearized heat semigroup and application of Li-Yau's inequalities.
result Established new Li-Yau's type inequalities for Finsler measure spaces.
The paper studies Harnack inequalities on Finsler metric measure spaces.
problem Analyzing Harnack inequalities on Finsler metric measure spaces.
method Using weighted Ricci curvature and distortion conditions, the authors derive an elliptic p-Harnack inequality.
result The paper establishes an elliptic p-Harnack inequality and derives Hölder continuity and gradient estimates for positive harmonic functions.
Proves inequality in metric measure spaces with non-branching structure.
problem Proving Heintze-Karcher inequality in metric measure spaces.
method Used needle decomposition technique for metric measure spaces.
result Characterizes equality case in spaces with positive curvature.
The paper derives concentration inequalities for dynamic risk measures in a Brownian filtration context.
problem Liquidity risk in financial markets.
method Backward stochastic differential equations (BSDEs) and their dual formulation.
result Derives concentration inequalities for time-consistent dynamic risk measures in a Brownian filtration.
Paper finds best constants in Hardy inequalities on Finsler metric measure manifolds.
problem Finding best constants in Hardy inequalities on Finsler metric measure manifolds.
method Investigates Hardy inequalities with distance functions in the Finsler setting, considering flag curvature, Ricci curvature, reversibility, and S-curvature.
result Establishes optimal Hardy inequalities on both noncompact and closed Finsler metric measure manifolds.
Study measures inequality in social-economic systems using Fokker-Planck equations and Lotka-Volterra dynamics.
problem Measuring inequality in oscillatory social-economic systems described by Fokker-Planck equations and Lotka-Volterra dynamics.
method Used Fokker-Planck equations and Lotka-Volterra dynamics to model inequality, focusing on coefficient of variation as a measure.
result Inequality initially tends to decrease in oscillatory systems, contrary to steady-state models.
Develops a framework for modeling liquidity risk using convex risk measures.
problem Modeling liquidity risk using convex risk measures.
method Exploits concentration of measure techniques to bound liquidity risk profiles.
result Derives tractable necessary and sufficient conditions for concentration inequalities of liquidity risk profiles.
The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.
problem Proving (p,q)-Sobolev and Nash inequalities on Finsler metric measure manifolds. method Global p-Poincaré inequality, (p,q)-Sobolev inequality, Nash inequality derivation. result Established global optimal (p,q)-Sobolev inequality with a sharp constant. We give sufficient conditions for a measured length space (X,d,m) to admit local and global Poincare inequalities. We first introduce a condition DM on (X,d,m), defined in terms of transport of measures. We show that DM, along with a doubling condition on m, implies a scale-invariant local Poincare inequality. We show …
Study Poincaré inequality in metric spaces via separating sets.
problem Geometric characterization of Poincaré inequality in metric spaces.
method Properties of separating sets and various notions of energy.
result Equivalence of conditions for 1-Poincaré inequality.
Generalizes Escobar-Riemann mapping problem for smooth metric measure spaces.
problem Finding a function that attains the Escobar weighted constant.
method Introducing Escobar quotient, infimum, and resolving the problem when the weighted constant is negative.
result Obtained an Aubin type inequality connecting weighted Escobar constant and optimal constant for trace inequality.
Sharp Poincaré inequality proved for specific metric spaces.
problem Proving a sharp Poincaré inequality for certain metric measure spaces.
method Identifying model densities and using localization arguments, without assuming geodesic convexity.
result Best possible Poincaré constant as a function of parameters.
We establish a Cauchy type inequality for the geometric intersection number between two 1-dimensional submanifolds in a surface. Some of the basic results in Thurston's theory of measured laminations on surfaces are derived from the Cauchy inequality.
Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincaré inequality", we provide a simple example of a metric space X that admits Poincaré inequalities for a continuum of mutually singular measures.
In this paper we present a correlation inequality with respect to Cauchy type measures. To prove our inequality, we transport the problem onto the Riemannian sphere then state and solve some special cases for a spherical correlation problem. This method, as we shall explain, opens up a new class of interesting problems…
Functional inequality proves quasi-invariance in infinite dimensions.
problem Proving quasi-invariance of measures in infinite-dimensional spaces.
method Using a functional inequality to prove quasi-invariance of measures under group actions.
result Different proof of the Cameron-Martin theorem in infinite dimensions.
Sharp isoperimetric inequalities for the sine transform of even isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. These new inequalities have direct applications to strong volume estimates for convex bodies from data about their sections or projec…
We define a Hamilton-Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1) a Talagrand inequality on a …
The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.
problem Rigidity of inequalities on weighted Riemannian manifolds.
method Theorems of rigidity on curvature and measure for the Borell-Brascamp-Lieb inequality, generalizing a theorem by Balogh and Kristály.
result A generalization of the curvature rigidity theorem to the weighted setting.
New PAC-Bayes bounds derived using Legendre transform and f-divergences.
problem Deriving PAC-Bayes bounds under various assumptions.
method Combining Legendre transform and Fenchel--Young inequality to derive change-of-measure inequalities.
result Extended PAC-Bayesian guarantees under tailored assumptions.
It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a…
A local cut point is by definition a point that disconnectes its sufficiently small neighborhood. We show that there exists an upper bound for the degree of a local cut point in a metric measure space satisfying the generalized Bishop--Gromov inequality. As a corollary, we obtain an upper bound for the number of ends o…
Method identifies low-dimensional structure in high-dimensional probability measures.
problem Identifying low-dimensional structure in high-dimensional probability measures.
method Extends prior work on minimizing majorizations of the Kullback-Leibler divergence to identify optimal approximations within a specific class of measures.
result Connection between dimensional logarithmic Sobolev inequality and approximations with the ansatz.
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. Paper finds inequalities for eigenvalues of buckling problems on special metric spaces.
problem Eigenvalue inequalities for buckling problems of drifting Laplacian.
method Investigated on bounded domains in complete smooth metric measure spaces (SMMSs) with special functions.
result General inequalities for eigenvalues derived under curvature constraints.
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 paper proves an inequality for symmetric polynomials under a fixed point measure.
problem An inequality for elementary symmetric polynomials under a fixed point measure of permutations.
method Constructing differential operators to set up a monotone flow.
result The inequality is proven and is sharp.
The hitting measure is singular and has dimension less than 1 for cocompact Fuchsian groups.
problem Analyzing the hitting measure and Hausdorff dimension for cocompact Fuchsian groups.
method Geometric and probabilistic analysis of random walks on cocompact Fuchsian groups.
result The hitting measure is singular with respect to Lebesgue measure and has a Hausdorff dimension strictly less than 1.
The paper proves a Moser-Trudinger inequality on metric spaces with curvature-dimension conditions.
problem Proving a Moser-Trudinger inequality on metric measure spaces.
method Rearrangement of functions on CD(k,n)-spaces satisfying a Polya-Szegö type inequality.
result Characterization of manifolds with lower bounded Ricci curvature admitting a Moser-Trudinger inequality.
Study examines how taxes affect wealth inequality in economic models.
problem Reducing economic inequality in models of economic activity.
method Examined Artificial Chemistry models and various tax measures.
result Effective tax measures can reduce economic inequality.
For certain metric spaces, Poincaré inequality does not improve even when spaces are blown up.
problem Improving Poincaré inequality with blow-up.
method Constructing families of metric measure spaces closed under blow-up.
result Poincaré inequality does not improve with blow-up for specific metric spaces.
Study inequality measures in wealth exchange models and compare with empirical data.
problem Analyzing inequality in wealth distribution models.
method Calculated Gini index and k-index, found bounds, and computed exact quantities for specific distributions.
result Found lower and upper bounds for inequality indices and discussed model efficiencies.
New equivalence found between curvature-dimension conditions and Wasserstein distance contraction.
problem Understanding the relationship between curvature-dimension conditions and Wasserstein distance contraction.
method Generalization of curvature-dimension conditions to metric measure spaces and proving equivalence with Wasserstein contraction properties.
result Wasserstein distance contraction properties are equivalent to curvature-dimension conditions.
The paper proves conditions for metric measure spaces to have specific volume growth and curvature properties.
problem Conditions for metric measure spaces to have specific volume growth and curvature properties.
method Proves conditions using volume doubling and Gagliardo-Nirenberg inequalities.
result Metric measure spaces with specific conditions have exactly the n-dimensional volume growth and zero flag curvature. 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.
Loewner inequality proven for curved surfaces.
problem Proving Loewner's inequality for nonpositively curved surfaces.
method Combining Gauss-Bonnet formula with averaging argument using geodesic flow invariance.
result Found a disk with large total curvature around its center, leading to large area.