Related papers: Cheeger type inequalities for high dimensional sim…
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…
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.
The Cheeger constant of a graph, or equivalently its coboundary expansion, quantifies the expansion of the graph. This notion assumes an implicit choice of a coefficient group, namely, $\mathbb{F}_2$. In this paper, we study Cheeger-type…
We construct two minimal Cheeger sets in the Euclidean plane, i.e. unique minimizers of the ratio "perimeter over area" among their own measurable subsets. The first one gives a counterexample to the so-called weak regularity property of…
In this paper we provide a framework for the study of isoperimetric problems in finitely generated group, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions,…
We use a variational formulation to define a generalized notion of perimeter, from which we derive abstract isoperimetric Cheeger's inequalities via gradient estimates on solutions of Poisson equations. Our abstract framework unifies many…
In this short exposition we provide a simplified proof of Buser's result for Cheeger's isoperimetric constant.
This is an expository article presenting various results about the Cheeger problem and its connections with other relevant mathematical problems. In section 2 we synthetically describe three mathematical problems (eigenvalue estimates,…
The Cheeger inequalities give an upper and lower bound on the spectral gap of discrete Laplacians defined on a graph in terms of the geometric characteristics of the graph. We generalise this approach and we employ it to determine if a…
We study several aspects of the $k$-th Cheeger constant of a complex X, a parameter that quantifies the distance of $X$ from a complex $Y$ with nontrivial $k$-th cohomology over $\mathbb{Z}_2$. Our results include general methods for…
This paper is devoted to a deep analysis of the process known as Cheeger deformation, applied to manifolds with isometric group actions. Here, we provide new curvature estimates near singular orbits and present several applications. As the…
In this paper we extend the known results of analytic connectivity to non-uniform hypergraphs. We prove a modified Cheeger's inequality and also give a bound on analytic connectivity with respect to the degree sequence and diameter of a…
We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more…
The Cheeger constant of an open set of the Euclidean space is defined by minimizing the ratio "perimeter over volume", among all its smooth compactly contained subsets. We consider a natural variant of this problem, where the volume of…
In this paper, we provide a lower bound for the Cheeger constant and the spectral gap for random complex curves in $\C P^2$. The complex curve is endowed with the restriction of the ambient Fubini-Study metric, and the probability measure…
In any dimension $n$, we determine the Cheeger constant and the Cheeger sets of the Gaussian mixture $\mu(x) = p\gamma(x-a) + (1-p)\gamma(x-b)$, where $p \in [0,1]$, $a,b \in \mathbb{R}^n$, and $\gamma : \mathbb{R}^n \to (0,\infty)$ denotes…
The higher-order Cheeger inequalities were established for graphs by Lee, Oveis Gharan and Trevisan. We prove analogous inequalities for graphons in this article.
The graph Cheeger constant and Cheeger inequalities are generalized to the case of hypergraphs whose edges have the same cardinality. In particular, it is shown that the second largest eigenvalue of the generalized normalized Laplacian is…
We show that under certain mild conditions, a metric simplicial complex which satisfies the Ptolemy inequality is a CAT(0) space. Ptolemy's inequality is closely related to inversions of metric spaces. For a large class of metric simplicial…
We establish in this note some Cauchy-Schwarz-type inequalities on compact K\"{a}hler manifolds, which generalize the classical Khovanskii-Teissier inequalities to higher-dimensional cases. Our proof is to make full use of the mixed…