```From:           asimov@nas.nasa.gov (Daniel A. Asimov)
Newsgroups:     sci.math.research
Subject:        Dense Sphere-Packings in Hyperbolic Space
Date:           11 Apr 1996 23:48:35 GMT
Organization:   NAS - NASA Ames Research Center, Moffett Field, CA
```

```(or maybe they should be called "ball-packings")

Consider hyperbolic space H^n (the unique smooth simply-connected Riemannian
n-manifold of constant sectional curvature = -1).

For each radius r > 0, one can ask the "Kepler question"

K(n,r):  What is the (or a) densest packing of H^n by balls of radius r ???

However, there are certain problems of well-definedness that arise:

Let Q(n,r) denote the set of all packings of H^n by closed balls of radius r
(where a packing is a collection of such balls any two of which are are either
disjoint or tangent).

For any q in Q(n,r) we let U(q) denote the union of all the balls of q, and
we let q(R; x) denote those points of U(q) that lie within a distance R
of the point x of H^n.

Given a packing q in Q(n,r), we can define the density d(q; x) as follows:

d(q; x) =   lim    vol(q(R; x)) / vol(B(R)),
R -> oo

if this limit exists, where B(R) denotes a ball of radius R in H^n.

-------------------------------------------------------------------------------
QUESTIONS:

1.	Given a packing q in Q(n,r), if d(q; x) exists for one x in H^n, does
it necessarily exist and have the same value for all x ???

Consider the packings q in Q(n,r) for which d(q; x) exists and is
independent of x.  (In this case we denote d(q; x) by d(q).)

Let supden(n,r) denote the supremum of all such d(q).

2.      Must there exist a packing q0 such that d(q0) = supden(n,r) ???

3.      Given n, is supden(n,r) a continuous function of r ???   Monotone ???

3.	Especially for n = 2 or n = 3, for which values of r is there an
explicit packing q in Q(n,r) that is known to realize supden(n,r) ???
-------------------------------------------------------------------------------

Daniel Asimov
Senior Research Scientist

Mail Stop T27A-1
NASA Ames Research Center
Moffett Field, CA 94035-1000

asimov@nas.nasa.gov
(415) 604-4799 w
(415) 604-3957 fax
```

```From:           kuperberg-greg@MATH.YALE.EDU (Greg Kuperberg)
Newsgroups:     sci.math.research
Subject:        Re: Dense Sphere-Packings in Hyperbolic Space
Date:           12 Apr 1996 15:12:09 -0400
Organization:   Yale University Mathematics Dept., New Haven, CT 06520-2155
```

```In article <4kk5oj\$ntl@cnn.nas.nasa.gov> asimov@nas.nasa.gov (Daniel A. Asimov) writes:
>1.	Given a packing q in Q(n,r), if d(q; x) exists for one x in H^n, does
>        it necessarily exist and have the same value for all x ???

[d(q;x) here is defined as the limit of the proportion covered
of round balls centered at x.]

Certainly not, given that in the hyperbolic plane, a cone with an angle
of 5 degrees contains a half-plane.

>Let supden(n,r) denote the supremum of all such d(q).
[assuming independence of x.]

>3.      Given n, is supden(n,r) a continuous function of r ???   Monotone ???

I doubt it, although I don't have a counterexample.

>4.	Especially for n = 2 or n = 3, for which values of r is there an
>	explicit packing q in Q(n,r) that is known to realize supden(n,r) ???