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Difference between revisions of "Deep hole"

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''in a lattice''
 
''in a lattice''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120080/d1200801.png" /> be a collection of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120080/d1200802.png" /> (usually a [[Lattice|lattice]]). A hole is a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120080/d1200803.png" /> whose distance to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120080/d1200804.png" /> is a local maximum. A deep hole is a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120080/d1200805.png" /> whose distance to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120080/d1200806.png" /> is the absolute maximum (if such exists).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120080/d1200807.png" /> is a lattice, then the holes are precisely the vertices of the Voronoi cells (cf. [[Voronoi diagram|Voronoi diagram]]; [[Parallelohedron|Parallelohedron]]).
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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
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article