arXiv research

In the class of smoothly embedded surfaces of sphere type we prove that the isoperimetric deficit can be controlled by the Willmore deficit.

Sharp bounds for curve isoperimetric deficit derived.

The paper improves inequalities for nearly spherical sets using quermassintegrals.

In this paper we provide a Bonnesen-style inequality which gives a lower bound for the isoperimetric deficit corresponding to a closed convex curve in terms of some geometrical invariants of this curve. Moreover we give a geometrical interpretation for the case when equality holds.

We relate the total curvature and the isoperimetric deficit of a curve $γ$ in a two-dimensional space of constant curvature with the area enclosed by the evolute of $γ$. We provide also a Gauss-Bonnet theorem for a special class of evolutes.

A sharp quantitative polygonal isoperimetric inequality is obtained.

The aim of this paper is to give not only an explicit upper bound of the total Q-curvature but also an induced isoperimetric deficit formula for the complete conformal metrics on $\mathbb R^n$, $n\ge 3$ with scalar curvature being nonnegative near infinity and Q-curvature being absolutely convergent.

Paper finds eigenvalue bounds for hyperbolic space domains.

We obtain a sharp lower bound on the isoperimetric deficit of a general polygon in terms of the variance of its side lengths, the variance of its radii, and its deviation from being convex. Our technique involves a functional minimization problem on a suitably constructed compact manifold and is based on the spectral t…

Quantifies fractional isoperimetric inequality with strong control over boundary oscillation.

The paper proves reverse inequalities in various geometric settings using curvature radius data.

We prove that finite perimeter subsets of $\mathbb{R}^{n+1}$ with small isoperimetric deficit have boundary Hausdorff-close to a sphere up to a subset of small measure. We also refine this closeness under some additional a priori integral curvature bounds. As an application, we answer a question raised by B. Colbois co…

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

Study stability of rigid motions and Möbius transformations on spheres, proving new rigidity estimates.

Unified approach to discrete and smooth isoperimetric inequalities of arbitrary order.

Study mass and center of mass in flat 3-manifolds, proving existence of foliations.

New proofs and inequalities for capillarity problems quantify asymmetries.

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…

On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius. The deficit between the diameters of the mani…

Optimally stabilizes Möbius group maps in spheres across dimensions.

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

New mass definition linked to ADM mass for general metrics.

Given a simple closed plane curve $Γ$ of length $L$ enclosing a compact convex set $K$ of area $F$, Hurwitz found an upper bound for the isoperimetric deficit, namely $L^2-4πF\leq π|F_{e}|$, where $F_{e}$ is the algebraic area enclosed by the evolute of $Γ$. In this note we improve this inequality finding strictly posi…

The study optimizes cell membranes' shapes based on curvature and proves existence of minimizers.

Kuwert and Schätzle showed in 2001 that the Willmore flow converges to a standard round sphere, if the initial energy is small. In this situation, we prove stability estimates for the barycenter and the quadratic moment of the surface. Moreover, in codimension one we obtain stability bounds for the enclosed volume and …

We generalise the classical Chern-Gauss-Bonnet formula to a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends results of Chang-Qing-Yang in the smooth case. Under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the si…

The purpose of this paper is to exhibit a quantitative stability result for the class of Möbius transformations of $\mathbb{S}^{n-1}$ when $n\geq 3$. The main estimate is of local nature and asserts that for a Lipschitz map that is apriori close to a Möbius transformation, an average conformal-isoperimetric type of def…

We are concerned about the coarse and precise aspects of a priori estimates for Green's function of a regular domain for the Laplacian-Betrami operator on any $3\le n$-dimensional complete non-compact boundary-free Riemannian manifold through the square Sobolev/Nash/logarithmic-Sobolev inequalities plus the rough and s…

In his paper "Shapes of Polyhedra and Triangulations of the Sphere", Thurston found that the set of shapes of convex polyhedra with prescribed cone-deficits has a complex hyperbolic structure. Inspired by his work, this paper studies the set of shapes of centrally symmetric octahedra with prescribed cone-deficits. We s…

Deep neural networks map brain lesions to deficits for better brain function understanding.

Study inequalities involving torsional rigidity and spherical deficit on Riemannian manifolds.

New deficit functions link elliptic and parabolic inequalities, proving log Sobolev.

Study analyzes household capital risk and poverty trapping, deriving a new function for capital deficit distribution.

The paper analyzes risk measures and optimal reserve allocation strategies.

This study analyzes public debts and deficits between European countries. The statistical evidence here seems in general to reveal that sovereign debts and government deficits of countries within European Monetary Unification-in average- are getting worse than countries outside European Monetary Unification, in particu…

We construct knot invariants on the basis of ascribing Euclidean geometric values to a triangulation of sphere S^3 where the knot lies. The main new feature of this construction compared to the author's earlier papers on manifold invariants is that now nonzero "deficit angles" (in the terminology of Regge calculus) can…

The accurate diagnosis and assessment of neurodegenerative disease and traumatic brain injuries (TBI) remain open challenges. Both cause cognitive and functional deficits due to focal axonal swellings (FAS), but it is difficult to deliver a prognosis due to our limited ability to assess damaged neurons at a cellular le…

Successful implementation of California's Renewable Portfolio Standard (RPS) mandating 33 percent renewable energy generation by 2020 requires inclusion of a robust strategy to mitigate increased risk of energy deficits (blackouts) due to short time-scale (sub 1 hour) intermittencies in renewable energy sources. Of the…

One of the main issues affecting the Italian NHS is the healthcare deficit: according to current agreements between the Italian State and its Regions, public funding of regional NHS is now limited to the amount of regional deficit and is subject to previous assessment of strict adherence to constraint on regional healt…

Quantitative estimates for $Q$-curvature near minimizing metrics on Riemannian manifolds.

We postulates, and then show experimentally, that liquidity deficit is the driving force of the markets. In the first part of the paper a kinematic of liquidity deficit is developed. The calculus-like approach, which is based on Radon--Nikodym derivatives and their generalization, allows us to calculate important chara…

New proof of log-Brunn-Minkowski inequality for zonoids and convex bodies.

Groups satisfy linear surface isoperimetric functions.

Solves relative isoperimetric problem on polygonal domains, focusing on corners.

The isoperimetric inequality and related inequalities are explored.

Isoperimetric regions in scaled product manifolds are products of regions in each factor.

The paper finds new inequalities for convex polygons.

Huisken's isoperimetric mass is always nonnegative.