Delaunay transformations of a Delaunay polytope

Laurent, Monique

Journal of Algebraic Combinatorics, Tome 5 (1996) no. 1, pp. 37-46.

Résumé

Summary: Let $P$ be a Delaunay polytope in $T( P )$ mathcalT$left( P right)$ denote the set of affine bijections $f$ of $T( P )$ {\mathcal{T}}$left( P \right)$ . We show that the dimension (in the topological sense) of the quotient set $T( P ) \mathord / \vphantom T( P )$     mathcalT$left( P right)$} \mathord{\left/ {$vphantom${{{\mathcal{T}}$left( P \right)} \sim $ right. kern-nulldelimiterspace $\sim $ coincides with another parameter of $P, namely,$ with its rank. Let $V$ denote the set of vertices of $P$ and let $d _{P}$ denote the distance on $V$ defined by $dp( u, u ) = || u$ - u || $^{2}$ dp$\left( {u,\upsilon } \right) = \left$| u - $\upsilon $ right|^2 for $u, v H _{| V |: = } { d$ | å $_{ u, n e V} b _{ u} b _{ n} d( u, n ) \leqslant 0$ mathcalH_left| V right|: = left{ left. d right|sumnolimits_u,nuvarepsilonV b_u $b_\nu d\left( {u,\nu } \right)} \leqslant 0}$ right. for b å $_{ u e V} b _{ u} = 1 }$ left. sumnolimits_u$\varepsilon V$ b_u = 1 right} . Then, the rank of $P$ is defined as the dimension of the smallest face of the cone $H _{| V |}$ mathcalH_left| V right| that contains $d _{P}$.

Détail

Classification : 11H06,52C07
Keywords: Delaunay polytope, affine transformation, lattice, dimension, hypermetric