01-395 Lewis Bowen

On the existence of completely saturated packings and completely reduced coverings (229K, postscript) Oct 24, 01

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Abstract. We prove the following conjecture of G. Fejes Toth, G. Kuperberg, and W. Kuperberg: every body $K$ in either $n$-dimensional Euclidean or $n$-dimensional hyperbolic space admits a completely saturated packing and a completely reduced covering. Also we prove the following counterintuitive result: for every $\epsilon > 0$, there is a body $K$ in hyperbolic $n$-space which admits a completely saturated packing with density less than $\epsilon$ but which also admits a tiling.

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