2015/05/14 by Sikiric, Mathieu Dutour
#FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1505.03687
For a lattice L of Rn, a sphere S(c,r) of center c and radius r is called \em empty if for any v∈ L we have \Vert v - c\Vert ≥ r. Then the set S(c,r)∩ L is the vertex set of a \em Delaunay polytope P=conv(S(c,r)∩ L). A Delaunay polytope is called \em perfect if any affine transformation ϕ such that ϕ(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 n≥ 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: 321 which is a Delaunay polytope in the root lattice E7 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.