Related papers: Minimal volume $k$-point lattice $d$-simplices

We consider random lattices taken from the general symplectic ensemble and count the number of lattice points of a typical lattice in nested families $B_t$ of certain Borel sets. Our main result is that for almost every general symplectic…

The conformal spectra of the critical dilute A-D-E lattice models are studied numerically. The results strongly indicate that, in branches 1 and 2, these models provide realizations of the complete A-D-E classification of unitary minimal…

Let $P$ be a bounded convex subset of $\mathbb R^n$ of positive volume. Denote the smallest degree of a polynomial $p(X_1,\dots,X_n)$ vanishing on $P\cap\mathbb Z^n$ by $r_P$ and denote the smallest number $u\geq0$ such that every function…

Given a degenerate $(n+1)$-simplex in a $d$-dimensional space $M^d$ (Euclidean, spherical or hyperbolic space, and $d\geq n$), for each $k$, $1\leq k\leq n$, Radon's theorem induces a partition of the set of $k$-faces into two subsets. We…

We find upper and lower bounds for the first eigenvalue and the volume entropy of a noncompact real analytic K\"ahler manifold, in terms of Calabi's diastasis function and diastatic entropy, which are sharp in the case of the complex…

It is shown that an n-dimensional unimodular lattice has minimal norm at most 2[n/24] +2, unless n = 23 when the bound must be increased by 1. This result was previously known only for even unimodular lattices. Quebbemann had extended the…

Let K and L be compact convex sets in R^n. The following two statements are shown to be equivalent: (i) For every polytope Q inside K having at most n+1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 <= d <=…

The Besson-Courtois-Gallot theorem is proven for noncompact finite volume Riemannian manifolds. In particular, no bounded geometry assumptions are made. This proves the minimal entropy conjecture for nonuniform rank one lattices.

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact…

It is classically known that closed geodesics on a compact Riemann surface with a metric of negative curvature strictly minimize length in their free homotopy class. We'd like to generalize this to Lagrangian submanifolds in K\"ahler…

In this paper we investigate the minimal and the next minimal volumes of normal KSBA stable surfaces with $p_g\ge 2$. We show that in case of $|K_X|$ not composed with a pencil, the minimal and next minimal volumes are $2p_g-4$ and…

In this note we show that the volume of axis-parallel boxes in $\mathbb{R}^d$ which do not intersect an admissible lattice $\mathbb{L}\subset\mathbb{R}^d$ is uniformly bounded. In particular, this implies that the dispersion of the dilated…

Let $\mathcal{E}$ be a set of points in $\mathbb{F}_q^d$. Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2016) proved that if $|\mathcal{E}|\gg q^{d-\frac{d-1}{k+1}}$ then $\mathcal{E}$ determines a positive proportion of all…

We develop a new framework to study minimum $d$-degree conditions in $k$-uniform hypergraphs, which guarantee the existence of a tight Hamilton cycle. Our main theoretical result deals with the typical absorption, path cover and connecting…

Let M be a compact manifold equipped with a Riemannian metric g and a spin structure \si. We let $\lambda (M,[g],\si)= \inf_{\tilde{g} \in [g]} \lambda_1^+(\tilde{g}) Vol(M,\tilde{g})^{1/n}$ where $\lambda_1^+(\tilde{g})$ is the smallest…

For a closed minimal submanifold $f:M^n\looparrowright \mathbb{S}^{N}$ in the unit sphere $(n<N)$, we prove $${\rm Vol}(M^n) \geq\frac{n+1}{n+2}\int_{M}\left( 1+\varphi_{p}^2\right) \geq m{\rm Vol}(\mathbb{S}^{n}),$$ where…

The Fine interior $F(P)$ of a $d$-dimensional lattice polytope $P \subset {\Bbb R}^d$ is the set of all points $y \in P$ having integral distance at least $1$ to any integral supporting hyperplane of $P$. We call a lattice polytope…

In a $d$-dimensional convex body $K$ random points $X_0, \dots, X_d$ are chosen. Their convex hull is a random simplex. The expected volume of a random simplex is monotone under set inclusion, if $K \subset L$ implies that the expected…

The groups of similarity and coincidence rotations of an arbitrary lattice L in d-dimensional Euclidean space are considered. It is shown that the group of similarity rotations contains the coincidence rotations as a normal subgroup.…

This article deals with topological assumptions under which the minimal volume entropy of a closed manifold $M$, and more generally of a finite simplicial complex $X$, vanishes or is positive. These topological conditions are expressed in…