Density of Tube Packings in Hyperbolic Space

Packing problems have long been a topic of interest. Traditionally, efforts had been focused on Euclidean space, but as interest in hyperbolic space has grown, many of the Euclidean problems have been translated into the hyperbolic arena, in which the problems are almost always vastly more complicated. The particular packing problem of interest here is a hyperbolic version of packing congruent right circular cylinders in Euclidean space. In Euclidean space, two equivalent ways to define a right circular cylinder are as the set of all points within a fixed distance of a given line or as the union of all lines passing perpendicularly through a given disk. In hyperbolic space, these two concepts are different. We will use the word tube in the former situation and the phrase right circular cylinder in the latter situation. Using this terminology, we are then investigating packings of congruent tubes in hyperbolic space. Density is perhaps the primary focus in any investigation of packings. Unfortunately, density can be somewhat difficult to define in hyperbolic space, especially when one is dealing with objects of infinite volume. We will simplify the issue by dealing with only a certain class of packings, although the result would likely follow in more general settings, assuming one defined density properly. Our main result is an upper bound on the density of symmetric packings of congruent tubes of radius r in hyperbolic space. We produce a means of computing the upper bound in arbitrary dimensions, and develop an explicit formula in dimension three. There is no reason to believe that our bounds are sharp, as we make a number of estimates along the way. We note that for the corresponding problem in three-dimensional Euclidean space, there is a sharp bound of π √ 12 [BK90]. In H3, there is a prior result [MM00a], which provides an upper bound for very large radius tubes and is asymptotically sharp. The result we develop here works well for moderate radius tubes.