Related papers: A reverse isoperimetric inequality for the Cheeger…
The $\mu$-conductance measure proposed by Lov\'asz and Simonovits is a size-specific conductance score that identifies the set with smallest conductance while disregarding those sets with volume smaller than a $\mu$ fraction of the whole…
A classical consequence of the John Ellipsoid Theorem is the upper bound $\sqrt{n}$ on the Banach-Mazur distance between the Euclidean ball and any symmetric convex body in $\mathbb{R}^n$. Equality is attained for the parallelotope and the…
In this paper, we prove a Poincar\'e-type inequality for any set of finite perimeter which is stable with respect to the free energy among volume-preserving perturbation, provided that the Hausdorff dimension of its singular set is at most…
We prove that the planar hexagonal honeycomb is asymptotically optimal for a large class of optimal partition problems, in which the cells are assumed to be convex, and the criterion is to minimize either the sum or the maximum among the…
The discrete Cheeger inequality, due to Alon and Milman (J. Comb. Theory Series B 1985), is an indispensable tool for converting the combinatorial condition of graph expansion to an algebraic condition on the eigenvalues of the graph…
Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in R^n. The corresponding inequality to the conjecture is sometimes called the the reverse…
This Thesis aims to highlight some isoperimetric questions involving the, so-called, $N$-clusters. We first briefly recall the theoretical framework we are adopting. This is done in Chapter one. In chapter two we focus on the standard…
The Falconer conjecture asserts that if E is a planar set with Hausdorff dimension strictly greater than 1, then its Euclidean distance set has positive one-dimensional Lebesgue measure. We discuss the analogous question with the Euclidean…
We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of curvature in discrete spaces. An appealing feature of this discrete version seems to be that it is fairly straightforward to compute this…
We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume of ellipsoids containing projections of convex bodies given…
Given an open and bounded set $\Omega\subset\mathbb{R}^N$, we consider the problem of minimizing the ratio between the $s-$perimeter and the $N-$dimensional Lebesgue measure among subsets of $\Omega$. This is the nonlocal version of the…
The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional)…
The Hausdorff-Young inequality for Euclidean space, in its sharp form due to Beckner, gives an upper bound for the Fourier transform in terms of Lebesgue space norms, with an optimal constant. The extremizers have been identified by Lieb to…
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 provide an upper bound on the Cheeger constant and first eigenvalue of the Laplacian of a finite-volume hyperbolic 3-manifold M, in terms of data from any surgery diagram for M. This has several consequences. We prove that a family of…
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…
Let M be a bounded domain of a Euclidian space with smooth boundary. We relate the Cheeger constant of M and the conductance of a neighborhood graph defined on a random sample from M. By restricting the minimization defining the latter over…
We prove that there are arbitrarily large equilateral sets of planar and symmetric convex bodies in the Banach--Mazur distance. The order of the size of these $d$-equilateral sets asymptotically matches the bounds of the size of…
We prove a sharp upper bound on convex domains, in terms of the diameter alone, of the best constant in a class of weighted Poincar\'e inequalities. The key point is the study of an optimal weighted Wirtinger inequality.
We study the Hausdorff distance between a set and its convex hull. Let $X$ be a Banach space, define the CHD-module of space $X$ as the supremum of this distance for all subset of the unit ball in $X$. In the case of finite dimensional…