from application tropical
Voronoi diagram with respect to the tropical metric in the tropical projective torus. Its combinatorics is controlled by a POLYTROPE_PARTITION
. See P. Criado, M. Joswig, P. Santos: Tropical bisectors and Voronoi diagrams, arXiv:1906.10950
AMBIENT_DIM
Number of dimensions of the diagram. One less than the number of coordinates.
N_SITES
Number of sites of the diagram.
POLYTROPE_PARTITION
Representation of the tropical Voronoi diagram. Each such polyhedron is a domain in which the distance to the set of sites $S$ is a minimum of linear functions. This list of regions is represented as an array of pairs of matrices. The first matrix in each pair represents the region itself (a polytrope) as a shortest path matrix. The second matrix (the labels) gives the index of the site $s\in S$ with maximum $s_j-s_i$ such that the cone $\{x:x_i-s_i<= x_k-s_k <= x_j-s_j \forall k\in [d+1]\}$ intersects this cell (or $-1$ if no such index exists). Then, in this region, $dist(x,S)$ is a minimum of the linear functions $(x_j-s_j)-(x_i-s_i)$ for each $s$ labelled with $(i,j)$.
Here is one polytrope cell.
> $T= new VoronoiDiagram(SITES=>[[-4,-4,0,0],[-3,0,2,0],[-2,-5,-2,0]]); > print $T->POLYTROPE_PARTITION->[0]; <0 inf inf inf -4 0 2 0 -5 inf 0 inf -4 inf inf 0 > <-1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 >
SITES
The sites of the tropical Voronoi diagram.