Related papers: The kissing number in four dimensions

A very fundamental geometric problem on finite systems of spheres was independently phrased by Kneser (1955) and Poulsen (1954). According to their well-known conjecture if a finite set of balls in Euclidean space is repositioned so that…

The Orbit Problem consists of determining, given a matrix $A\in \mathbb{R}^{d\times d}$ and vectors $x,y\in \mathbb{R}^d$, whether there exists $n\in \mathbb{N}$ such that $A^n=y$. This problem was shown to be decidable in a seminal work of…

Every graph $G$ can be represented by a collection of equi-radii spheres in a $d$-dimensional metric $\Delta$ such that there is an edge $uv$ in $G$ if and only if the spheres corresponding to $u$ and $v$ intersect. The smallest integer $d$…

We apply the semidefinite programming approach developed in arxiv:math.MG/0608426 to obtain new upper bounds for codes in spherical caps. We compute new upper bounds for the one-sided kissing number in several dimensions where we in…

A well-known chessboard problem is that of placing eight queens on the chessboard so that no two queens are able to attack each other. (Recall that a queen can attack anything on the same row, column, or diagonal as itself.) This problem is…

We give the general Kerr-de Sitter metric in arbitrary spacetime dimension D\ge 4, with the maximal number [(D-1)/2] of independent rotation parameters. We obtain the metric in Kerr-Schild form, where it is written as the sum of a de Sitter…

This is the sixth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…

Littlewood asked for the maximum number $N$ of congruent infinite cylinders that can be arranged in $\mathbb{R}^3$ so that every pair touches. We improve upon the proof of the second author that $N \leq 18$ to show that $N \leq 10$.…

Let $B$ be a set of $n$ axis-parallel boxes in $\mathbb{R}^d$ such that each box has a corner at the origin and the other corner in the positive quadrant of $\mathbb{R}^d$, and let $k$ be a positive integer. We study the problem of…

In 1933, Borsuk made a conjecture that every $n$-dimensional bounded set can be divided into $n+1$ subsets of smaller diameter. Up to now, the problem is still open for $4\leq n\leq 63$. In this paper, we firstly discuss the Banach-Mazur…

The problem of finding the asymptotic behavior of the maximal density of sphere packings in high Euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry. One century ago, Minkowski obtained a…

We show that on an $n=24m+8k$-dimensional even unimodular lattice, if the shortest vector length is $\geq 2m$, then as the number of vectors of length $2m$ decreases, the secrecy gain increases. We will also prove a similar result on…

We prove by example that the number of elastic collisions of $n$ balls of equal mass and equal size in $d$-dimensional space can be greater than $n^3/27$ for $n\geq 3$ and $d\geq 2$. The previously known lower bound was of order $n^2$.

In the papers Ziegler(2001) and Goldstein(2012) it was previously shown that any subset of the Boolean cube $ S \subset \{0,1\}^n $ for $ n \leq 9 $ can be partitioned into $n+1$ parts of smaller diameter, i.e., the Borsuk conjecture holds…

A lattice $(d,k)$-polytope is the convex hull of a set of points in $\mathbb{R}^d$ whose coordinates are integers ranging between $0$ and $k$. We consider the smallest possible distance $\varepsilon(d,k)$ between two disjoint lattice…

The contact number of a packing of finitely many balls in Euclidean $d$-space is the number of touching pairs of balls in the packing. A prominent subfamily of sphere packings is formed by the so-called totally separable sphere packings:…

In the last decade, copositive formulations have been proposed for a variety of combinatorial optimization problems, for example the stability number (independence number). In this paper, we generalize this approach to infinite graphs and…

Let's have $n$ points in the space such that the maximum distance between any of them is $a$. We prove that there exists a sphere of radius $r \leq a \frac{\sqrt(6)}{4}$ that contains in its interior or on its surface all these points.…

An open problem that is widely regarded as one of the most important in quantum query complexity is to resolve the quantum query complexity of the k-distinctness function on inputs of size N. While the case of k=2 (also called Element…

For given positive integers $m$ and $n$ with $m<n$, the Prouhet-Tarry-Escott problem asks if there exist two disjoint multisets of integers of size $n$ having identical $k$th moments for $1\leq k\leq m$; in the ideal case one requires…