Related papers: The KLS Isoperimetric Conjecture for Generalized O…
We reinterpret the renormalized volume as the asymptotic difference of the isoperimetric profiles for convex co-compact hyperbolic 3-manifolds. By similar techniques we also prove a sharp Minkowski inequality for horospherically convex sets…
We give a new proof of the isoperimetric inequality in the plane, based on Steiner's formula for the area of a convex neighborhood. This proof establishes the isoperimetric inequality directly, without requiring that we separately establish…
We reconsider the spinfoam dynamics that has been recently introduced, in the generalized Kaminski-Kisielowski-Lewandowski (KKL) version where the foam is not dual to a triangulation. We study the Euclidean as well as the Lorentzian case.…
We study the way in which the Euclidean subspaces of a Banach space fit together, somewhat in the spirit of the Ka\v{s}in decomposition. The main tool that we introduce is an estimate regarding the convex hull of a convex body in John's…
Let $K\subset \mathbb{R}^n$ be a convex body, $n\geq 3$. We say that $K$ satisfies the Barker-Larman condition if there exists a ball $B$ in the interior of $K$ such that for every suppor hyperplane $\Pi$ of $B$, the section $\Pi \cap K$ is…
In this paper, we obtain the best possible value of the absolute constant $C$ such that for every isotropic convex body $K \subseteq \mathbb{R}^n$ the following inequality (which was proved by Klartag and reduces the hyperplane conjecture…
We establish geometric upper and lower estimates for the Carath\'eodory and Kobayashi-Eisenman volume elements on the class of non-degenerate convex domains, as well as on the more general class of non-degenerate $\mathbb{C}$-convex…
We show that as the ratio between the first Dirichlet eigenvalues of a convex domain and of the ball with the same volume becomes large, the same must happen to the corresponding ratio of isoperimetric constants. The proof is based on the…
The Cohn-Kumar conjecture states that the triangular lattice in dimension 2, the $E_8$ lattice in dimension 8, and the Leech lattice in dimension 24 are universally minimizing in the sense that they minimize the total pair interaction…
In this paper we prove that among all convex domains of the plane with two axis of symmetry, the maximizer of the first non trivial Neumann eigenvalue $\mu_1$ with perimeter constraint is achieved by the square and the equilateral triangle.…
In this paper, we consider a finite-dimensional optimization problem minimizing a continuous objective on a compact domain subject to a multi-dimensional constraint function. For the latter, we assume the availability of a global Lipschitz…
According to Suk's breakthrough result on the Erdos-Szekeres problem, any point set in general position in the plane, which has no $n$ elements that form the vertex set of a convex $n$-gon, has at most $2^{n+O\left({n^{2/3}\log n}\right)}$…
Given a knot in 3-space, one can associate a sequence of Laurrent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The Generalized Volume Conjecture states that the value of the $n$-th colored Jones polynomial at $\exp(2…
General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the boundary of the geometric (Minkowski) sum of $k$ ellipsoids in $n$-dimensional Euclidean space. Previously this was done…
General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the boundary of the geometric (Minkowski) sum of $k$ ellipsoids in $n$-dimensional Euclidean space. Previously this was done…
In this survey, we discuss volumetric and combinatorial results concerning (mostly finite) intersections or unions of balls (mostly of equal radii) in the $d$-dimensional real vector space, mostly equipped with the Euclidean norm. Our first…
In this work we discuss a conjecture of Viterbo relating the symplectic capacity of a convex body and its volume. The conjecture states that among all 2n-dimensional convex bodies with a given volume the euclidean ball has maximal…
In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H. Poincar\'{e} Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman studied low-dimensional projections of random vectors from the Euclidean unit sphere and…
For any origin-symmetric convex body $K$ in $\mathbb{R}^n$ in isotropic position, we obtain the bound: \[ M^*(K) \leq C \sqrt{n} \log(n)^2 L_K ~, \] where $M^*(K)$ denotes (half) the mean-width of $K$, $L_K$ is the isotropic constant of…
We find an optimal upper bound on the volume of the John ellipsoid of a $k$-dimensional section of the $n$-dimensional cube, and an optimal lower bound on the volume of the L\"owner ellipsoid of a projection of the $n$-dimensional…