arXiv research

Quantifies fractional isoperimetric inequality with strong control over boundary oscillation.

Study shows how close functions are to optimal in Riemannian manifolds.

We study the Riemannian quantiative isoperimetric inequality. We show that direct analogue of the Euclidean quantitative isoperimetric inequality is--in general--false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) versio…

The paper proves stability of inequalities for nearly spherical sets in various spaces.

New proofs and inequalities for capillarity problems quantify asymmetries.

The paper develops quantitative estimates for holomorphic sections over bounded domains.

New proof of Willmore inequality using geometric divergence inequality.

We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with $\mathrm{Ric}_{\infty} \ge 1$. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle deco…

The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdré). Then we give a quan…

Sharp 3D Alexandrov inequality applied to volume-preserving flows.

The paper improves inequalities for nearly spherical sets using quermassintegrals.

The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.

Sharp inequalities and symmetries on Riemannian surfaces quantified.

Quantifies nearly spherical subsets in complex ball geometry.

Researchers prove a stability result for a 3-sphere inequality, extending previous work.

Study sharp inequalities for perimeter functionals in capillarity and convex cones.

The paper provides precise estimates for isoperimetric inequalities on weighted manifolds.

Sharp inequality proven for symmetric functions on a 4D sphere.

Optimally estimates stability in Lorentzian isoperimetric inequalities.

A sharp quantitative polygonal isoperimetric inequality is obtained.

Improved Yang-Yau inequality for all orientable surfaces except for specific genera.

Explains optimal functional inequalities, focusing on Sobolev and fractional Sobolev.

Sharp stability of Möbius group among sphere-valued maps proved in arbitrary dimensions.

This paper uses second-order Poincaré inequalities to establish quantitative central limit theorems for Gaussian neural networks.

In this paper, we study the sharp constants of quantitative Hardy and Rellich inequalities on nonreversible Finsler manifolds equipped with arbitrary measures. In particular, these inequalities can be globally refined by adding remainder terms like the Brezis-Vázquez improvement, if Finsler manifolds are of strictly ne…

The paper extends intersection theory for b-divisors, proving monotonicity and volume inequalities.

Proves a quantitative index theorem for positive scalar curvature metrics.

The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.

Using the method of De Lellis-Topping, we prove some almost Schur type results. For example, one of our results gives a quantitative measure of how close the higher mean curvature of a submanifold is to its average value. We also derive another sharp Andrews-De Lellis-Topping type inequality involving the Riemannian cu…

Sharp Talenti-type comparison theorem for p-Laplacian on RCD(K,N) spaces.

New inequality for eigenfunctions on curved spaces.

The study extends isoperimetric inequalities to non-positive curvature spaces.

We obtain the best known quantitative estimates for the $L^p$-Poincaré and log-Sobolev inequalities on domains in various sub-Riemannian manifolds, including ideal Carnot groups and in particular ideal generalized H-type Carnot groups and the Heisenberg groups, corank $1$ Carnot groups, the Grushin plane, and various H…

The paper extends geometric inequalities for nearly spherical sets in various space forms.

Proves inequality linking function deviation to gradient norm on compact manifolds.

Adapts Stein's method for geometric inequalities, addressing boundary terms.

We prove that the log-Brunn-Minkowski inequality \begin{equation*} |λK+_0 (1-λ)L|\geq |K|^λ|L|^{1-λ} \end{equation*} (where $|\cdot|$ is the Lebesgue measure and $+_0$ is the so-called log-addition) holds when $K\subset\mathbb{R}^n$ is a ball and $L$ is a symmetric convex body in a suitable $C^2$ neighborhood of $K$.

The paper proves and analyzes Minkowski inequalities for nearly spherical domains.

We point out a simple equities trading strategy that allows a sufficiently large, market-neutral, quantitative hedge fund to achieve outsized returns while simultaneously contributing significantly to increasing global wealth inequality. Overnight and intraday return distributions in major equity indices in the United …

Short note proves Poincaré inequality for 4-manifold forms.

In this paper we revisit the anisotropic isoperimetric and the Brunn-Minkowski inequalities for convex sets. The best known constant $C(n)=Cn^{7}$ depending on the space dimension $n$ in both inequalities is due to Segal [\ref{bib:Seg.}]. We improve that constant to $Cn^6$ for convex sets and to $Cn^5$ for centrally sy…

In a seminal paper "Volumen und Oberfläche" (1903), Minkowski introduced the basic notion of mixed volumes and the corresponding inequalities that lie at the heart of convex geometry. The fundamental importance of characterizing the extremals of these inequalities was already emphasized by Minkowski himself, but has to…

Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.

The paper proves inequalities for star-shaped and $F$-mean convex hypersurfaces in $\mathbb{R}^{n+1}$.

New method uses entropy dissipation to prove isoperimetric inequalities.

In this paper we extend the results of "A strong minimax property of nondegenerate minimal submanifolds" by White, where it is proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in a certain geodesic tubular neighbourhood…

Study on stability of free boundary Willmore problem using new gradient inequality.

Bounds neural network output distribution to Gaussian for random initialization.