Commit e77549a2 authored by Saul Schleimer's avatar Saul Schleimer

Added discussion of dimension.

parent 6447ee8a
2018-09-07 - or a few days earlier
2018-09-07
Note that there are easy upper bounds on the number of facets
(codimension one faces) of the core and geometric polytopes. To see
this: Cut down the parameter space using the equalities, to get a
single linear subspace of some dimension. If the triangulation is
core then this dimension is already the stratum dimension (as all
variables can be positive). Now cut down using the positivity
constraints - we get at most one facet for every positivity
constraint. Thus the number of facets of the core polytope is at most
twice the number of edges: 12g - 12 + 6p.
[This should be reduced to the number of small branches in the
vertical track plus the number of small branches in the horizontal
track, as the positivity of one of these implies positivity of the
edges "downstream" until the sink. Experiments show that there can be
dependencies between small branches, so the number of faces can be
smaller even smaller... ]
The geometric polytope is the same dimension (by assumption!) so the
number of additional facets is at most the number of flippable edges
(forward or backward). This is at most two-thirds the number of
edges, so we get 4g - 4 + 2p more giving an upper bound of
16g - 16 + 8p
faces. Note that the Keane faces where already counted previously,
and all faces are either Keane or Delaunay. Note furthermore that the
number of faces of any linear subspace intersected with the geometric
polytope has at most this number of facets as well.
2018-09-04 - or a few days earlier
1. Compute the shape of the octagon surface. Guess the linear space.
Check it has the correct dimension. Carry it around and see if the
......
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