Related papers: Empty axis-parallel boxes

In dimension $d$, Mutually Unbiased Bases (MUBs) are a collection of orthonormal bases over $\mathbb{C}^d$ such that for any two vectors $v_1, v_2$ belonging to different bases, the scalar product $|\braket{v_1|v_2}| = \frac{1}{\sqrt{d}}$.…

A definition of asymptotic flatness at spatial infinity in $d$ dimensions ($d\geq 4$) is given using the conformal completion approach. Then we discuss asymptotic symmetry and conserved quantities. As in four dimensions, in $d$ dimensions…

Geometrical objects with integral sides have attracted mathematicians for ages. For example, the problem to prove or to disprove the existence of a perfect box, that is, a rectangular parallelepiped with all edges, face diagonals and space…

We prove the uniqueness theorem for asymptotically flat static vacuum black hole solutions in higher dimensional space-times. We also construct infinitely many non-asymptotically flat regular static black holes on the same spacetime…

Let $d \geq 1$ and $s \leq 2^d$ be nonnegative integers. For a subset $A$ of vertices of the hypercube $Q_n$ and $n\geq d$, let $\lambda(n,d,s,A)$ denote the fraction of subcubes $Q_d$ of $Q_n$ that contain exactly $s$ vertices of $A$. Let…

I present analytic time symmetric initial data for five dimensions describing ``bubbles of nothing'' which are asymptotically flat in the higher dimensional sense, i.e. there is no Kaluza-Klein circle asymptotically. The mass and size of…

An axis-parallel $b$-dimensional box is a Cartesian product $R_1\times R_2\times...\times R_b$ where $R_i$ is a closed interval of the form $[a_i,b_i]$ on the real line. For a graph $G$, its \emph{boxicity} box(G) is the minimum dimension…

The Generalized Lax Conjecture asks whether every hyperbolicity cone is a section of a semidefinite cone of sufficiently high dimension. We prove that the space of hyperbolicity cones of hyperbolic polynomials of degree $d$ in $n$ variables…

This paper studies cross-sections inside black holes and conjectures a universal inequality: in a static $(d+1)$-dimensional asymptotically planar/spherical Schwarzschild-AdS spacetime of given energy $E$ and AdS radius $\ell_{\text{AdS}}$,…

An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328k^2 points in the plane contains an empty…

This paper considers the question of how to succinctly approximate a multidimensional convex body by a polytope. Given a convex body $K$ of unit diameter in Euclidean $d$-dimensional space (where $d$ is a constant) and an error parameter…

Let $S$ be a set of $n$ points in the unit square $[0,1]^2$, one of which is the origin. We construct $n$ pairwise interior-disjoint axis-aligned empty rectangles such that the lower left corner of each rectangle is a point in $S$, and the…

The Loomis-Whitney inequality, and the more general Uniform Cover inequality, bound the volume of a body in terms of a product of the volumes of lower-dimensional projections of the body. In this paper, we prove stability versions of these…

We obtain an optimal upper bound for the normalised volume of a hyperplane section of an origin-symmetric d-dimensional cube. This confirms a conjecture posed by Imre Barany and Peter Frankl.

Static, spherically symmetric solutions of the field equations for a particular dimensional continuation of general relativity with negative cosmological constant are studied. The action is, in odd dimensions, the Chern-Simons form for the…

We study the $(n-1)$-dimensional volume of central hyperplane sections of the $n$-dimensional cube $Q_n$. Our main goal is two-fold: first, we provide an alternative, simpler argument for proving that the volume of the section perpendicular…

Let $P_{n}$ be a set of $n$ points, including the origin, in the unit square $U = [0,1]^2$. We consider the problem of constructing $n$ axis-parallel and mutually disjoint rectangles inside $U$ such that the bottom-left corner of each…

We give an algebraic characterization of when a $d$-dimensional periodic framework has no non-trivial, symmetry preserving, motion for any choice of periodicity lattice. Our condition is decidable, and we provide a simple algorithm that…

We introduce a minor variant of the approximate D-optimal design of experiments with a more general information matrix that takes into account the representation of the design space S. The main motivation (and result) is that if S in R^d is…

Let $M^d$ denote the $d$-dimensional Euclidean, hyperbolic, or spherical space. The $r$-dual set of given set in $M^d$ is the intersection of balls of radii $r$ centered at the points of the given set. In this paper we prove that for any…