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- W4298142520 abstract "For a lattice $L$ of $R^n$, a sphere $S(c,r)$ of center $c$ and radius $r$ is called {em empty} if for any $vin L$ we have $Vert v - cVert geq r$. Then the set $S(c,r)cap L$ is the vertex set of a {em Delaunay polytope} $P=conv(S(c,r)cap L)$. A Delaunay polytope is called {em perfect} if any affine transformation $phi$ such that $phi(P)$ is a Delaunay polytope is necessarily an isometry of the space composed with an homothety. Perfect Delaunay polytopes are remarkable structure that exist only if $n=1$ or $ngeq 6$ and they have shown up recently in covering maxima studies. Here we give a general algorithm for their enumeration that relies on the Erdahl cone. We apply this algorithm in dimension 7 which allow us to find that there are only two perfect Delaunay polytopes: $3_{21}$ which is a Delaunay polytope in the root lattice $mathsf{E}_7$ and the Erdahl Rybnikov polytope. We then use this classification in order to get the list of all types Delaunay simplices in dimension 7 and found 11 types." @default.
- W4298142520 created "2022-10-01" @default.
- W4298142520 creator A5032090330 @default.
- W4298142520 date "2015-05-14" @default.
- W4298142520 modified "2023-09-26" @default.
- W4298142520 title "The seven dimensional perfect Delaunay polytopes and Delaunay simplices" @default.
- W4298142520 doi "https://doi.org/10.48550/arxiv.1505.03687" @default.
- W4298142520 hasPublicationYear "2015" @default.
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