Search Results

You are looking at 1 - 5 of 5 items for :

Clear All

Abstract  

We construct a convex body K ⊃ ℝ3 such that the maximum number of mutually nonoverlapping translates of K which touch K is 15.

Restricted access

Abstract  

We give a very short survey of the results on placing of points into the unit n-dimensional cube with mutual distances at least one. The main result is that into the 5-dimensional unit cube there can be placed no more than 40 points.

Restricted access

Abstract  

We present a very short survey of known results and many new estimates and results on the maximum number of points that can be chosen in the n-dimensional unit cube so that every distance between them is at least 1.

Restricted access

Abstract

The aim of this paper is to determine the locally densest horoball packing arrangements and their densities with respect to fully asymptotic tetrahedra with at least one plane of symmetry in hyperbolic 3-space extended with its absolute figure, where the ideal centers of horoballs give rise to vertices of a fully asymptotic tetrahedron. We allow horoballs of different types at the various vertices. Moreover, we generalize the notion of the simplicial density function in the extended hyperbolic space (n≧2), and prove that, in this sense, the well known Böröczky–Florian density upper bound for “congruent horoball” packings of does not remain valid to the fully asymptotic tetrahedra.

The density of this locally densest packing is ≈0.874994, may be surprisingly larger than the Böröczky–Florian density upper bound ≈0.853276 but our local ball arrangement seems not to have extension to the whole hyperbolic space.

Restricted access
Authors: Károly Bezdek and Robert Connelly

Abstract  

Suppose that p = (p1, p2, …, pN) and q = (q1, q2, …, qN) are two configurations in

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{E}^d$$ \end{document}
, which are centers of balls Bd(pi, r i) and Bd(qi, r i) of radius r i, for i = 1, …, N. In [9] it was conjectured that if the pairwise distances between ball centers p are contracted in going to the centers q, then the volume of the union of the balls does not increase. For d = 2 this was proved in [1], and for the case when the centers are contracted continuously for all d in [2]. One extension of the Kneser-Poulsen conjecture, suggested in [6], was to consider various Boolean expressions in the unions and intersections of the balls, called flowers, where appropriate pairs of centers are only permitted to increase, and others are only permitted to decrease. Again under these distance constraints, the volume of the flower was conjectured to change in a monotone way. Here we show that these generalized Kneser-Poulsen flower conjectures are equivalent to an inequality between certain integrals of functions (called flower weight functions) over
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{E}^d$$ \end{document}
, where the functions in question are constructed from maximum and minimum operations applied to functions each being radially symmetric monotone decreasing and integrable.

Restricted access