Paper proves Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
problem Proving equivalence between Brunn-Minkowski inequality and curvature dimension condition.
method Analyzes weighted Riemannian manifolds, proving equivalence without optimal transport or differential structure.
result Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
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.
The paper proves new inequalities for convex hypersurfaces using centro-affine geometry.
problem Proving inequalities for convex hypersurfaces.
method Introducing a flat logarithmic centro-affine geometry and using Bochner formulas.
result Established new Poincaré and Brunn-Minkowski inequalities.
Paper generalizes Minkowski inequality for umbilical hypersurfaces with free boundary.
problem Optimal weighted Minkowski inequality for umbilical hypersurfaces with free boundary.
method Generalization of Xia's result to free boundary case in space forms.
result Free boundary version of optimal weighted Minkowski inequality in space forms.
The paper proves new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
problem Proving new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
method Locally constrained inverse curvature flows in hyperbolic and spherical spaces.
result Established new Alexandrov-Fenchel and Minkowski inequalities involving general convex weight functions.
New weighted surface area measures for convex bodies with applications.
problem Generalizing surface area measures to weighted Borel measures.
method Formulating and analyzing weighted surface area measures, proving integral formula and Bézout-type inequality.
result New integral formula for mixed measure of three bodies, generalizing Bézout-type inequality.
Sharp Minkowski inequality found for AdS-Melvin spacetime surfaces.
problem Proving a Minkowski-type inequality for surfaces in the AdS-Melvin space.
method Used weighted normal flow to prove inequality for general surfaces.
result Sharp Minkowski inequality holds for all surfaces in AdS-Melvin space.
Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.
problem Deriving upper bounds for the Alexandrov-Fenchel deficit.
method Using weighted Minkowski integral formulas and an integral formula for the deficit in Jensen's inequality.
result Quantitative estimates under weaker convexity assumptions, including a distance term.
Extends global stability of Minkowski spacetime to minimal decay assumptions.
problem Global stability of Minkowski spacetime under minimal decay assumptions.
method Uses rp-weighted estimates instead of vectorfield method. result Proves global stability of Minkowski spacetime under minimal decay assumptions.
New proof of Minkowski spacetime stability in exterior regions.
problem Stability of Minkowski spacetime in exterior regions.
method Unified treatment of decay of initial data, use of rp-weighted estimates. result Reduced number of derivatives and simplified last slice treatment.
The traditional Minkowski distances are induced by the corresponding Minkowski norms in real-valued vector spaces. In this work, we propose novel statistical symmetric distances based on the Minkowski's inequality for probability densities belonging to Lebesgue spaces. These statistical Minkowski distances admit closed…
We use the weighted Hsiung-Minkowski integral formulas and Brendle's inequality to show new rigidity results. First, we prove Alexandrov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a large class of Riemannian warped product manifolds, including the Schwarzschild…
Paper proves rigidity results for anisotropic capillary hypersurfaces.
problem Rigidity of anisotropic capillary hypersurfaces.
method New Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces.
result Uniqueness of solution to anisotropic Orlicz-Christoffel-Minkowski problem.
New inequalities for convex hypersurfaces in various spaces.
problem Deriving inequalities for hypersurfaces under convex weight.
method Sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces in Euclidean, spherical, and hyperbolic spaces.
result Incorporates convex, non-decreasing positive functions as weights, yielding a broad family of geometric inequalities.
We study a Riemannian manifold equipped with a density which satisfies the Bakry--Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincaré-type inequality on its boundary assuming that the latter is local…
We prove a generalization of Hsiung-Minkowski formulas for closed submanifolds in semi-Riemannian manifolds with constant curvature. As a corollary, we obtain volume and area upper bounds for k-convex hypersurfaces in terms of a weighted total k-th mean curvature of the hypersurface. We also obtain some Alexandrov-type…
Proposes a new K-means method for efficient clustering of nonlinear data.
problem Challenges of kernel K-means, including high memory usage and computational inefficiency.
method Combines linear and nonlinear approaches using explicit feature maps based on spectral analysis.
result Demonstrates Explicit Kernel Minkowski Weighted K-means (Explicit KMWK-means) reduces memory usage and improves efficiency.
This paper classifies solutions to a specific hyperbolic geometry problem.
problem Classifying solutions to a specific hyperbolic geometry equation.
method Analytical and numerical methods to solve the equation.
result Classification of solutions for p≥−7, nonuniqueness for p<−7. New insights from centro-affine geometry solve a key geometric conjecture.
problem Log-Brunn-Minkowski conjecture in centro-affine differential geometry.
method Interpreting the log-Brunn-Minkowski conjecture as a spectral problem and using centro-affine differential geometry.
result Global uniqueness and inequalities in the log-Minkowski problem for certain convex bodies.
Proves stability of Minkowski space-time for Einstein-Yang-Mills equations.
problem Stability of Minkowski space-time for perturbations governed by Einstein-Yang-Mills equations.
method Proves exterior energy estimates for tensorial non-linear wave equations in Minkowski space-time.
result Proves exterior stability of Minkowski space-time for Einstein-Yang-Mills equations.
In this paper, the radiation field is defined for solutions to Einstein vacuum equations which are close to Minkowski space-time with spacial dimension n≥4. The regularity properties and asymptotic behavior of those Einstein vacuum solutions are established at the same time. In particular, the map from Cauchy int…
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…
Minkowski space is shown to be globally stable as a solution to the Einstein--Vlasov system in the case when all particles have zero mass. The proof proceeds by showing that the matter must be supported in the "wave zone", and then proving a small data semi-global existence result for the characteristic initial value p…
The paper extends geometric inequalities for nearly spherical sets in various space forms.
problem Investigating weighted inequalities for nearly spherical sets in space forms.
method Generalizing and extending inequalities for nearly spherical sets in C1 and W2,∞ settings, with convex weight functions. result Quantitative stability estimates for weighted inequalities in Rn+1 and Hn+1. In this paper, we use the inverse mean curvature flow to establish an optimal Minkowski type inquality, weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Reissner-Nordström-anti-deSitter manifold and Penrose type inequality for asymptotically locally hyperbolic manifolds in which c…
Böröczky, Lutwak, Yang and Zhang recently proved the log-Brunn-Minkowski inequality which is stronger than the classical Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper establishes the log-Brunn-Minkowski, log-Minkowski, Lp-Minkowski and Lp-Brunn-Minkowski inequalities f…
New curvature-dimension condition for Lagrangians on manifolds.
problem Establishing a curvature-dimension condition for autonomous Lagrangians.
method Generalizing Klartag's needle decomposition technique to Lagrangian setting.
result Equivalence of curvature-dimension condition to displacement convexity of entropy.
The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.
problem Solving the (p,q)-Christoffel-Minkowski problem.
method Investigating the problem via an expanding curvature flow.
result Existence and uniqueness of smooth solutions to the (p,q)-Christoffel-Minkowski problem.
We consider inverse curvature flows in warped product manifolds, which are constrained subject to local terms of lower order, namely the radial coordinate and the generalized support function. Under various assumptions we prove longtime existence and smooth convergence to a coordinate slice. We apply this result to ded…
Study anisotropic flows solving Orlicz-Minkowski problems, proving existence and new results.
problem Anisotropic non-homogeneous Gauss curvature flows and Orlicz-Minkowski problems.
method Long-time existence and behavior analysis, parabolic approximation method, curvature flow.
result Existence and new results for Orlicz-Minkowski problems, including Lp versions. Optimizes transport in Finsler spacetimes with lower Ricci bounds.
problem Optimizing transport in Finsler spacetimes with lower Ricci bounds.
method Using optimal transport and weighted Ricci curvature bounds.
result Proves timelike curvature-dimension condition for Finsler spacetimes.
The paper generalizes Alexandrov theorems for null hypersurfaces with integral curvature conditions.
problem Determining when a submanifold lies on a shear-free null hypersurface under integral curvature conditions.
method Using Minkowski formulas with arbitrary weight to derive rigidity results for submanifolds with weaker integral curvature conditions.
result A necessary and sufficient condition for a submanifold to lie in a shear-free null hypersurface is given by a mean curvature integral inequality.
Researchers prove uniqueness and continuity of solution to L_p dual Minkowski problem.
problem Proving uniqueness and continuity of solution to L_p dual Minkowski problem.
method Established new Minkowski-type inequalities related to optimization problem.
result Uniqueness and continuity of solution for general convex bodies when q<p. Developed a sub-Riemannian version of Minkowski problem in Heisenberg groups.
problem Minkowski type problem in Heisenberg groups.
method Variational method.
result Positive answer to sub-Riemannian Minkowski type problem.
Paper solves Minkowski problem for anisotropic p-torsional rigidity.
problem Solving the Minkowski problem for anisotropic p-torsional rigidity.
method Using the anisotropic p-Laplacian equation, presenting sufficient and necessary conditions for existence. result Presented sufficient and necessary conditions for the existence of a solution.
Develops Brunn-Minkowski theory in hyperbolic space using hyperbolic p-sum.
problem No universally accepted sum for sets in hyperbolic space.
method Introduces hyperbolic p-sum and develops horospherical p-Brunn-Minkowski theory.
result Solves p-Minkowski and p-Christoffel-Minkowski problems for various p.
Paper proves anisotropic Minkowski inequality and related inequalities.
problem Proving anisotropic Minkowski inequality and related inequalities.
method Utilizes a nonlinear potential theoretic approach.
result Sharp anisotropic Minkowski inequality and related inequalities proved.
Characterizes Lorentzian manifolds embeddable in Minkowski spacetime.
problem Identifying Lorentzian manifolds embeddable in Minkowski spacetime.
method Characterization and proof of embeddability conditions.
result Lorentzian manifolds embeddable in Minkowski spacetime coincide with globally hyperbolic spacetimes.
Magnetic Brunn-Minkowski inequalities on Riemannian manifolds
problem Establishing equivalence between Brunn-Minkowski inequalities and magnetic Ricci curvature
method Using magnetic geodesics
result Proving a sharp, undistorted Brunn-Minkowski inequality
Paper solves capillary Orlicz-Minkowski problem with new inequalities.
problem Finding capillary convex bodies with prescribed Orlicz surface area measures.
method Continuity method and inequalities to solve the capillary even Orlicz-Minkowski problem.
result Volume-normalized smooth solutions and inequalities established.
Solves a discrete logarithmic Minkowski problem for electrostatic p-capacity.
problem Characterize measures generated by electrostatic p-capacity.
method Solves the discrete logarithmic Minkowski problem for 1 < p < n.
result Solves the discrete logarithmic Minkowski problem for measures in general position.
New method solves generalized Minkowski problem for torsional rigidity.
problem Generalized Minkowski problem for torsional rigidity.
method Flow method
result Existence of solutions for general measures.
Paper solves Minkowski problem for p-harmonic measures.
problem Solving the Minkowski problem for p-harmonic measures on convex domains.
method Using the Gauss curvature flow method.
result Existence of smooth solution to the Minkowski problem for p-harmonic measures.
Study anisotropic inverse Gauss curvature flows and solve dual Orlicz Minkowski problems.
problem Solving dual Orlicz Minkowski problems for anisotropic flows.
method Anisotropic inverse Gauss curvature flows and stationary solutions.
result New existence results for dual Orlicz Minkowski problems for smooth measures.
Paper solves Minkowski problem for non-compact convex sets with asymptotic boundary conditions.
problem Solving Minkowski problem for non-compact convex sets with asymptotic boundary conditions.
method Combining covolume, Hadamard variational formula, and geometric interpretation.
result Solved Minkowski problem for non-compact convex sets under asymptotic conditions.
Researchers solved the even Lp-Minkowski problem under curvature pinching.
problem Solving the even Lp-Minkowski problem under curvature pinching. method Anisotropic Riemannian metric comparison and anisotropic curvature analysis.
result The even Lp-Minkowski inequality and uniqueness are proven for all p≥pγ. Solves a generalized dual Minkowski problem for specific values of q.
problem Finding solutions to the generalized dual Minkowski problem for given q and star bodies.
method Variational methods
result Existence of solutions for q<0 and 0≤q≤1, sufficient condition for q>1.
Proves equality in Minkowski inequality for static, flat manifolds.
problem Proving equality in Minkowski inequality for static, flat manifolds.
method Analyzes quasi-spherical metrics and static manifolds.
result Equality in Minkowski inequality achieved only by Schwarzschild space slices.