and Projections of Polytopes
Dissertation (Advisor: Günter M. Ziegler, TU Berlin)
The main reason for the fact that triangulations of cyclic polytopes can be treated effectively in a purely combinatorial way are the following well-known properties that follow from the special structure of Vandermonde-determinants.
The first one -- Gale's famous Evenness Criterion -- characterizes the set of all combinatorial facets of . The following notion allows us to state that criterion in a compact way.
The second one describes the form of those sets of vertices of whose convex hulls intersect in the relative interior of both. Hence, this determines .
The combinatorial polytopes are identical for all because the strictly monotone function does not affect the assertions of these criteria. This means that the combinatorial study of triangulations of cyclic polytopes with any parametrization is equivalent to the investigation of combinatorial triangulations of the combinatorial polytopes .
The set of circuits with maximal element in is denoted by , the set of circuits having their maximal element in is written as . The cyclic polytope labelled by is denoted by .
Note that in odd dimensions there are polytopes that have the same face lattice as but a different circuit structure (see ); this leads to completely different triangulations.
The upper facets of are the sets of those facets of that can be seen from a point in with large positive -th coordinate (geometric upper facets of ), the lower facets label the sets of those facets of that can be seen from a point in with large negative -th coordinate (geometric lower facets of ). The geometric upper (respectively lower) facets project down to without overlapping. Therefore, their projections define geometric triangulations of .
The support of any circuit in corresponds to the label set of a unique -simplex in where its set of geometric upper facets belongs to the elements of the star of the positive part in , and its set of geometric lower facets corresponds to the elements of the star of the negative part in .
If the rows are filled with stars corresponding to two simplices
then these two simplices are admissible if and only if each zig-zag-path
connects at most
stars. For instance,
the table looks as
The reader will easily find a zig-zag-path connecting even stars, showing that is not an admissible pair.
Obviously all with are isomorphic to . From now on we are exclusively dealing with combinatorial triangulations of , and we will leave out the ``combinatorial'' attribute whenever this is not confusing.
The following Propositions -- consequences of Theorems 3.3.3 and 3.3.4 -- relate cyclic polytopes with different parameters. We use the notation for and for .
The following proposition is the combinatorial description for the geometric connection provided by the projection between -simplices in and the minimal affine dependencies in .