View source for Deep hole

''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).

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]]).

====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>