Sharp stability results for reverse isoperimetric inequalities in 2D.
problem Reverse isoperimetric inequalities in the plane.
method Stability analysis of λ-convex bodies and convex bodies with smooth boundaries. result Sharp stability results for reverse isoperimetric inequalities, including inradius and Cheeger inequalities.
The paper examines stability of Minkowski inequalities in warped product spaces.
problem Stability of Minkowski-type inequalities for hypersurfaces in warped product spaces.
method Established a stability estimate for the norm of the traceless second fundamental form.
result Proved stability of Minkowski inequalities in specific warped product examples.
The paper examines the stability of Minkowski inequality for nearly spherical domains.
problem Stability of Minkowski inequality for nearly spherical domains.
method Analyzes stability inequalities for C1 perturbations of a ball and axially symmetric perturbations. result Established stability inequalities for curvature integrals of nearly spherical domains.
Paper proves stability of quermassintegral inequalities using inverse curvature flow.
problem Stability of quermassintegral inequalities for nearly spherical sets.
method Inverse curvature flow with special rescaling to study quermassintegral inequalities.
result Decreasing rate of k-th quermassintegral is faster than Fraenkel asymmetry for nearly spherical sets.
Optimally estimates stability in Lorentzian isoperimetric inequalities.
problem Stability estimates in Lorentzian isoperimetric inequalities.
method Quantitative stability estimates using Fraenkel asymmetry and Lipschitz bounds.
result Optimal stability estimates with universal constants for Lorentzian isoperimetric inequalities.
The paper proves stability of inequalities for nearly spherical sets in various spaces.
problem Stability of geometric inequalities for nearly spherical sets.
method Deriving a quantitative quermassintegral inequality and applying it to derive stability results.
result Stability of geometric inequalities involving weighted curvature integrals and quermassintegrals for nearly spherical sets in Rn+1 and Hn+1. The study examines stability of Sobolev inequalities on manifolds with Ricci bounds.
problem Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.
method Combines techniques from smooth and non-smooth geometry, focusing on direct strategies.
result Effective methods revealed for stability of Sobolev inequalities on manifolds with non-negative Ricci curvature and Euclidean volume growth.
The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.
problem Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.
method Generalized Lions' concentration compactness and rigidity results of Sobolev inequalities on singular spaces.
result Almost extremal functions are close to extremal functions on the round sphere and Euclidean Sobolev inequality.
Explains optimal functional inequalities, focusing on Sobolev and fractional Sobolev.
problem Optimal functional inequalities and their stability.
method Compactness theorems, characterization of optimizers, and quantitative stability analysis.
result Characterization and stability of optimizers for Sobolev inequalities and their fractional generalizations.
We give a stability version of of the Blaschke-Santaló inequality in the plane.
New inequality helps map stability in minimal surfaces.
problem Stability of minimal surfaces in Rn. method Developing new inequalities and perspectives on minimal surfaces.
result Reproves instability of classical minimal surfaces like Enneper.
Adapts Stein's method for geometric inequalities, addressing boundary terms.
problem Geometric inequalities and their stability under constraints.
method Uses elliptic PDE with oblique boundary condition to handle boundary terms.
result Stability results for various geometric inequalities with respect to a new distance.
Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
problem Stability of the Schwarzschild 3-manifold in the context of the 3D Riemannian Penrose inequality.
method Pointed measured Gromov-Hausdorff topology, negligible domains and boundary area perturbations.
result Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.
problem Investigating stability of the Sobolev inequality in metric spaces with curvature dimension conditions.
method Assuming almost the same optimal constant, the paper shows that the cumulative distribution of almost extremal functions is close to that of an Aubin-Talenti bubble on the round sphere.
result Quantitative stability with sharp exponent for the Sobolev inequality in various curvature and dimension assumptions.
The paper proves LOO CV is reliable under estimator stability.
problem Ensuring the reliability of leave-one-out cross validation.
method Using concentration inequalities based on logarithmic Sobolev inequality.
result LOO CV is a valid procedure under estimator stability.
Researchers prove a stability result for a 3-sphere inequality, extending previous work.
problem Quantitative stability of nonlinear Yamabe-type inequalities on the 3-sphere.
method Proved a two-term refinement of the Schur lemma inequality in the conformal class of the 3-sphere.
result Deduced quantitative stability of an entire family of nonlinear Yamabe-type inequalities.
Stability inequalities for specific solutions in high dimensions.
problem Stability of solutions to the one-phase Bernoulli problem.
method Proving strict stability inequalities for cohomogeneity one solutions with bi-orthogonal symmetry.
result Strict stability for cohomogeneity one solutions in dimensions 7 and above.
The study proves stability of quermassintegral inequalities in hyperbolic space.
problem Stability of quermassintegral inequalities for horospherically convex hypersurfaces in hyperbolic space.
method Using initial value independent curvature estimates for locally constrained flows of inverse type.
result Explicit exponent of the deficit in the quermassintegral inequality is given and does not depend on dimension.
Uniqueness of nondegenerate blowups for planar networks shown.
problem Uniqueness of nondegenerate blowups for the motion by curvature of planar networks.
method Proof based on Lojasiewicz-Simon gradient inequality applied to stability properties of critical points of the length functional.
result Uniqueness of nondegenerate compact blowups for the motion by curvature of planar networks.
The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.
problem Deriving new inequalities on manifolds and convex hypersurfaces.
method Using Fourier theory and geometric implications of Poincare-type inequalities.
result Sharp Minkowski-type inequalities, including stability and Alexandrov-Fenchel inequalities.
Study shows how close functions are to optimal in Riemannian manifolds.
problem Understanding how close functions are to optimal in Riemannian manifolds.
method Analyzes quantitative stability of Sobolev inequalities on compact Riemannian manifolds.
result Functions that nearly saturate a critical Sobolev inequality are quantitatively close to extremal functions.
The paper proves stability of eigenvalue inequalities on surfaces.
problem Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces.
method Employing eigenvalues of measures and Sobolev space W−1,2, the paper proves stability estimates for the first and second nonzero Laplace eigenvalues on surfaces. result Metrics almost maximizing the normalized eigenvalue are W−1,2-close to a maximal metric. New approach to concentration inequalities for unbounded state space dynamical systems.
problem Concentration inequalities for unbounded state space dynamical systems.
method Functional analytic framework, transport-entropy inequality.
result Exponential concentration inequalities for sampling from stationary distribution.
Study on stability of free boundary Willmore problem using new gradient inequality.
problem Stability of free boundary Willmore problem.
method New Łojasiewicz-Simon gradient inequality for functionals on infinite dimensional manifolds.
result Existence and convergence of solutions for the free boundary Willmore flow.
The paper proves geometric inequalities and their stabilities for curves in hyperbolic space.
problem Geometric inequalities and their stabilities for curves in hyperbolic space.
method Curve flow for shifted principal curvatures, Heintze-Karcher type inequality for h-convex curves.
result Geometric inequalities and their stabilities for curves in hyperbolic space.
In this paper, we deals with isoperimetric-type inequalities for closed convex curves in the Euclidean plane R^2. We derive a family of parametric inequalities involving the following geometric functionals associated to a given convex curve with a simple Fourier series proof: length, area of the region included by the …
Study on stability of ALE Ricci-flat metrics using a modified Perelman's λ-functional.
problem Stability and instability of ALE Ricci-flat metrics.
method Use of a modified Perelman's λ-functional and Lojasiewicz inequality.
result Demonstrates dynamical instability of ALE Ricci-flat metrics.
The paper proves stability and convergence of minimal networks under curvature motion.
problem Stability and convergence of minimal networks under curvature motion.
method Proved Lojasiewicz-Simon gradient inequalities for minimal networks.
result Motion by curvature starting from networks close to minimal ones exists for all times and smoothly converges.
We prove stability inequalities for Lawson cones Mkh with (k,h),(h,k)∈{(3,5),(2,7),(2,8),(2,9),(2,10),(2,11)}. This extends the results of G. De. Philippis and F. Maggi to all area-minimizing Lawson cones.
We introduce the notions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. We prove that for any projective K3 surface, there exists a constant C depending only on the rank and discriminant of its Picard group, such that $$\mathrm{sys}(σ)^2\leq C\cdot\mathrm{…
The study proves stability of the positive mass theorem for Kähler manifolds.
problem Stability of the positive mass theorem for Kähler manifolds.
method Integral inequality and stability results for ADM mass on AE Kähler manifolds.
result Stability of the positive mass theorem for Kähler manifolds under certain conditions.
Refines spinorial Sobolev inequality on sphere, proving stability and new properties of Killing spinors.
problem Stability of spinorial Sobolev inequalities on the unit sphere.
method Establishes a refined spinorial Sobolev inequality and proves stability.
result Stability inequality and new properties of Killing spinors.
Extends fractional Lp uncertainty principles with extremizers and stability results.
problem Investigating uncertainty principles in fractional Lp settings. method Analyzing the fractional Schrödinger equation to find extremal functions and sharp constants.
result Proves stability of extremizers for fractional uncertainty inequalities.
We show that the cone-volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition. This implies, among others, a conjectured best possible inequality for the U-functional of a convex body. For both results we provide stronger versions in the sense of stability i…
Introduces a universal Bochner formula for scalar curvature.
problem None explicitly stated; focuses on a new formula.
method Introduces a universal Bochner formula.
result Contains special cases like stability inequality and Schrödinger-Lichnerowicz-type formula.
Study stability of surfaces in spacetimes, proving new estimates and theorems.
problem Stability of surfaces in spacetime and their applications.
method Variational techniques, Christodoulou-Yau estimate, Cohn-Vossen inequality, global theorem, capillary stability, area inequality, diameter estimate.
result Established new estimates and theorems for stable surfaces in spacetime.
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. The stability of the Yamabe invariant of S3 is discussed.
problem Stability of the Yamabe invariant of S3. method Analysis of asymptotically flat metrics with vanishing scalar curvature and nearly Euclidean L2 Sobolev inequality. result If a manifold (R3,g) carries a suitably normalized, positive solution to Δgw+λw5=0, then w must be close to a conformal factor transforming Euclidean space into a round sphere. We prove that the elliptic Harnack inequality (on a manifold, graph, or suitably regular metric measure space) is stable under bounded perturbations, as well as rough isometries.
Proves stability of spacetime Penrose inequality for spherical symmetric initial data.
problem Stability of the Penrose inequality for spherical symmetric spacetimes.
method Formulated and proved stability statement using spherical symmetry and asymptotically flat initial data.
result Initial data must arise from an isometric embedding into a static spacetime close to Schwarzschild spacetime.
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.
The study improves Poincaré and log-Sobolev inequalities on hyperbolic spaces.
problem Improving Poincaré and log-Sobolev inequalities on hyperbolic spaces.
method Establishing scale-dependent Poincaré-Hardy type identities and choosing suitable parameters, potentials, and vector fields.
result Derives new versions and substantially improves existing inequalities.
P. Berard and D. Meyer proved a Faber-Krahn inequality for domains in compact manifolds with positive Ricci curvature. We prove stability results for this inequality.
The paper proves stability of Wulff shapes using anisotropic curvature functionals.
problem Stability of Wulff shapes under anisotropic curvature.
method Estimates distance to Wulff shape using Lp-norm of traceless F-Hessian of a foliating function. result Quantitative stability results for anisotropic inequalities and problems.
Market makers optimize trading with a new implicit scheme for complex inequalities.
problem Optimizing trading in a limit order book with stochastic and impulse control.
method Implicit numerical scheme coupled with policy iteration algorithm.
result Convergence to the unique viscosity solution of the HJBQVI.
The article analyzes the stability of a curve shortening flow for planar networks.
problem Stability analysis of anisotropic curve shortening flow for planar networks.
method Used Lojasiewicz-Simon gradient inequality to derive stability results.
result For initial data close to an energy minimizer, the flow exists globally and converges to a different energy minimum.
New stability framework relaxes boundedness assumptions for generalization bounds.
problem Overly restrictive assumptions for modern learning settings with heavy-tailed or unbounded losses.
method Develops a stability-based framework requiring only finite Lp moment conditions. result Sharp generalization bounds derived for various learning paradigms.
We present three ways to establish general stability inequalities for various classes of 2-immersions in Euclidean spaces of higher codimension