arXiv research

Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.

Sharp stability results for reverse isoperimetric inequalities in 2D.

The paper characterizes hyperbolic manifolds and graphs verifying a specific isoperimetric inequality.

Quantifies fractional isoperimetric inequality with strong control over boundary oscillation.

In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we…

Study shows convergence rates for Cheeger cuts on data clouds.

We prove a Cheeger inequality for the first positive Steklov eigenvalue. It involves two isoperimetric constants.

New proofs of Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.

The paper studies isoperimetric inequalities on warped product manifolds.

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.

The study finds a special isoperimetric inequality for minimal hypersurfaces in spheres.

Negative curvature manifolds have vanishing bounded volume class if and only if Cheeger constant is positive.

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…

We prove that the results regarding the Isoperimetric inequality and Cheeger constant formulated in terms of the Minkowski content, obtained by the authors in previous papers in the framework of essentially non-branching metric measure spaces verifying the local curvature dimension condition, also hold in the stronger …

Paper connects probability density cuts to graph theory eigenfunctions.

We show that the Cheeger constant of compact surfaces is bounded by a function of the area. We apply this to isoperimetric profiles of bounded genus non-compact surfaces, to show that if their isoperimetric profile grows faster than $\sqrt t$, then it grows at least as fast as a linear function. This generalizes a resu…

Simplified proof for Cheeger's isoperimetric constant.

We give a brief literature review of the isoperimetric problem and discuss its relationship with the Cheeger constant of Riemannian $n$-manifolds. For some non-compact, finite area 2-manifolds, we prove the existence and regularity of subsets whose isoperimetric ratio is equal to the Cheeger constant. To do this, we us…

We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative Bakry-Émery Ricci curvature. We derive some universal inequalities among eigenvalues of the weighted Laplacian on such manifolds. These inequalities are quantitative versions of the previous t…

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…

The goal of the paper is to sharpen and generalise bounds involving the Cheeger's isoperimetric constant $h$ and the first eigenvalue $λ_{1}$ of the Laplacian. A celebrated lower bound of $λ_{1}$ in terms of $h$, $λ_{1}\geq h^{2}/4$, was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on $λ_{1…

We obtain an estimate of the Cheeger isoperimetric constant in terms of the volume growth for a properly immersed submanifold in a Riemannian manifold which possesses at least one pole and sectional curvature bounded from above .

The paper proves a diastolic inequality linking surface area and loop length.

The Cheeger constant increases under Ricci flow on spheres.

We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative, and more generally, is in the range $N \in (-\infty,1)$, extending the scope from the traditional range $N \i…

We extend several Cheeger-type isoperimetric bounds for convex sets in Euclidean space, due to Bobkov and Kannan-Lovász-Simonovits, to Riemannian manifolds having non-negative Ricci curvature. In order to extend Bobkov's bound, we require in addition an upper bound on the sectional curvature of the space, which permits…

Study on Cheeger constant in specific hyperbolic 3-manifolds.

We discuss the behavior of $(λ_{1. p}(M))^{1/p}$ with respect to the Gromov-Hausdorff topology and the variable $p$, where $λ_{1, p}(M)$ is the first positive eigenvalue of the $p$-Laplacian on a compact Riemannian manifold $M$. Applications include new estimates for the first eigenvalues of the $p$-Laplacian on Rieman…

We show that the Cheeger isoperimetric constant of a solvable simply connected Lie group $G$ with Lie algebra $\G$ is $h(G)=\max_{H\in\G,||H||=1} \tr(\ad (H))$.

New inequality for special forms on manifolds.

The isoperimetric inequality and related inequalities are explored.

The paper finds new inequalities for convex polygons.

The paper proves inequalities for Steklov eigenvalues on finite graphs.

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a compl…

Solved Cheeger inequalities for simplicial complexes, combining topological and graph theoretic methods.

We introduce notions of Cheeger constants for graphons and graphings. We prove Cheeger and Buser inequalities for these. On the way we prove co-area formulae for graphons and graphings.

Lower bounds for eigenvalues on manifolds with boundary conditions.

The paper proves new inequalities in hyperbolic space using Euclidean methods.

Study Cheeger inequalities for Riemannian manifolds with boundary.

Study improves isoperimetric inequality for random groups.

The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.

Proves optimal isoperimetric inequality in de Sitter space.

Sharp isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature.

Solves equality case in isoperimetric inequality for non-convex domains.

Generalizes Cheeger inequality to Carnot-Carathéodory spaces.

Prove an isoperimetric inequality for compact bodies in 3D contact non-unimodular Lie groups.

New findings on metric spaces with finite Nagata dimension.

Proof of reverse isoperimetric inequality for black holes.