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Deep hole

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in a lattice

Let $P$ be a collection of points in ${\mathbf R}^n$ (usually a lattice of points). A hole is a point of ${\mathbf R}^n$ whose distance to $P$ is a local maximum. A deep hole is a point of ${\mathbf R}^n$ whose distance to $P$ is the absolute maximum (if such exists).

If $P$ is a lattice, then the holes are precisely the vertices of the Voronoi cells (cf. Voronoi diagram; Parallelohedron).

References

[a1] J.H. Conway, N.J.A. Sloane, "Sphere packings, lattices and groups" , Grundlehren , 230 , Springer (1988) pp. 6; 26; 33; 407
How to Cite This Entry:
Deep hole. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Deep_hole&oldid=45669
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article