Difference between revisions of "Deep hole"
From Encyclopedia of Mathematics
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| − | Let | + | 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 | + | If $P$ is a lattice, then the holes are precisely the vertices of the Voronoi cells (cf. [[Voronoi diagram|Voronoi diagram]]; [[Parallelohedron|Parallelohedron]]). |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.H. Conway, N.J.A. Sloane, "Sphere packings, lattices and groups" , ''Grundlehren'' , '''230''' , Springer (1988) pp. 6; 26; 33; 407</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.H. Conway, N.J.A. Sloane, "Sphere packings, lattices and groups" , ''Grundlehren'' , '''230''' , Springer (1988) pp. 6; 26; 33; 407</TD></TR></table> | ||
Latest revision as of 21:02, 2 May 2020
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=12811
Deep hole. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Deep_hole&oldid=12811
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article