Related papers: A Blaschke-Lebesgue Theorem for the Cheeger consta…

Let $d \geq 2$. The Cheeger constant of a graph is the minimum surface-to-volume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and volume here: the simplest method is to…

We obtain all extreme and exposed points of the closed unit ball of the space of bilinear forms $T:\ell_{\infty}^{2}\times\ell_{\infty}^{2}\rightarrow \mathbb{R}.$ We also show that any (norm one) bilinear form $T:\ell_{\infty…

Brooks and Makover developed a combinatorial model of random hyperbolic surfaces by gluing certain hyperbolic ideal triangles. In this paper we show that for any $\epsilon>0$, as the number of ideal triangles goes to infinity, a generic…

The object of the paper is to find complete systems of inequalities relating the perimeter $P$, the area $|\cdot|$ and the Cheeger constant $h$ of planar sets. To do so, we study the so called Blaschke--Santal\'o diagram of the triplet…

In this paper we consider the scale invariant shape functional $${\mathcal{F}}_{p,q}(\Omega)=\frac{\lambda_p^{1/p}(\Omega)}{\lambda_q^{1/q}(\Omega)},$$ where $1\le q<p\le+\infty$ and $\lambda_p(\Omega)$ (respectively $\lambda_q(\Omega)$) is…

We prove a Payne-Weinberger type inequality for the $p$-Laplacian Neumann eigenvalues ($p\ge 2$). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constants in Poincar\'e…

We study the Cheeger isoperimetric constant as a functional in the space of convex co-compact hyperbolic 3-manifolds which are either quasi-Fuchsian or acylindrical. We prove that the global maximum is attained uniquely at the Fuchsian…

We prove a lower bound for the Cheeger constant of a cylinder $\Omega\times (0,L)$, where $\Omega$ is an open and bounded set. As a consequence, we obtain existence of minimizers for the shape functional defined as the ratio between the…

In this paper we introduce a Cheeger-type constant defined as a minimization of a suitable functional among all the $N$-clusters contained in an open bounded set $\Omega$. Here with $N$-Cluster we mean a family of $N$ sets of finite…

This note details how a recent structure theorem for normal $1$-currents proved by the first and third author allows to prove a conjecture of Cheeger concerning the structure of Lipschitz differentiability spaces. More precisely, we show…

We show how 'test' vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we…

Lebesgue space bounds $L^{p_1}({\mathbb R}^1) \times L^{p_2}(^1) \to L^q({\mathbb R}^1)$ are established for certain maximal bilinear operators. The proof combines a trilinear smoothing inequality with Calder\'on-Zygmund theory. A reference…

We generalize the Cheeger inequality, a lower bound on the first nontrivial eigenvalue of a Laplacian, to the case of geometric sub-Laplacians on rank-varying Carnot-Carath\'eodory spaces and we describe a concrete method to lower bound the…

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…

This paper investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly weighted and/or with boundary. The higher Cheeger constants…

We investigate numerically the optimal constants in Lieb-Thirring inequalities by studying the associated maximization problem. We use a monotonic fixed-point algorithm and a finite element discretization to obtain trial potentials which…

We study the optimal constant in a Sobolev inequality for BV functions with zero mean value and vanishing outside a bounded open set. We are interested in finding the best possible embedding constant in terms of the measure of the domain…

We present a new, computer-assisted, proof that for all triangles in the plane, the equilateral triangle uniquely maximizes the ratio of the first two Dirichlet-Laplacian eigenvalues. This proves an independent proof the triangular…

We prove that, in the limit as $k \to+ \infty$, the hexagonal honeycomb solves the optimal partition problem in which the criterion is minimizing the largest among the Cheeger constants of $k$ mutually disjoint cells in a planar domain. As…

We prove that the first eigenvalue of the Dirichlet Laplacian for a triangle in the plane is bounded above by $\pi^2 L^2\over 9A^2$, where $L$ is the perimeter and $A$ is the area of this triangle. We show that the \mbox{constant 9} is…