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.
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.
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 ofdoes 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.
, which are centers of balls Bd(pi, ri) and Bd(qi, ri) of radius ri, for i = 1, …, N. In  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 , and for the case when the centers are contracted continuously for all d in . One extension
of the Kneser-Poulsen conjecture, suggested in , 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