===== A Counter-example to an integer analog to Caratheodory's Theorem =====

==== The construction ====

This tutorial describes the construction of a specific rational cone in six dimensions which is due to:

  * Bruns, Winfried; Gubeladze, Joseph; Henk, Martin; Martin, Alexander; Weismantel, Robert: A counterexample to an integer analogue of Carathéodory's theorem. J. Reine Angew. Math. 510 (1999), 179-185.

The rows of this matrix describe a cone //C//:

<code perl>
> $M = new Matrix<Rational>([[0,1,0,0,0,0],
> [0,0,1,0,0,0],
> [0,0,0,1,0,0],
> [0,0,0,0,1,0],
> [0,0,0,0,0,1],
> [1,0,2,1,1,2],
> [1,2,0,2,1,1],
> [1,1,2,0,2,1],
> [1,1,1,2,0,2],
> [1,2,1,1,2,0]]);
> $C=new Polytope<Rational>(POINTS=>$M);
</code>
From

<code perl>
> print $C->HILBERT_BASIS;
0 0 0 0 0 1
0 0 0 0 1 0
0 0 0 1 0 0
0 0 1 0 0 0
1 0 2 1 1 2
0 1 0 0 0 0
1 1 1 2 0 2
1 1 2 0 2 1
1 2 0 2 1 1
1 2 1 1 2 0

</code>
one can see that the given generators of //C// form a Hilbert basis. Now we consider one particular point //x//. The output of the second command (all coefficients positive) shows that //x// is contained in the interior of //C//.

<code perl>
> $x=new Vector<Rational>([9,13,13,13,13,13]);
> print $C->FACETS * $x;
8 15 19/2 19/2 17 13 17 13 9 13 13 17 8 19/2 13 17 15 19/2 15 15 19/2 17 11 15 8 8 8

</code>
The following loop iterates over all invertible 6x6 submatrices of //M// and computes the unique representation of //x// as a linear combination of the rows of the submatrix. The output (suppressed as it is too long) shows that each such linear combination requires at least one negative or one non-integral coefficient.

<code perl>
> foreach (@{all_subsets_of_k(range(0,9),6)}) {
>   $B = $M->minor($_,All);
>   if (det($B)) {
>     print lin_solve(transpose($B),$x), "\n";
>   }
> }
</code>
This means that //x// cannot be represented as a non-negative linear combination of any six of the given generators of //C//.

==== Analyzing the combinatorics ====

The following is taken from

  * Michael Joswig, Benjamin Müller, and Andreas Paffenholz: ''polymake'' and lattice polytopes. In Christian Krattenthaler, Volker Strehl and Manuel Kauers (eds.), Proceedings of the 21th International Conference on Formal Power Series and Algebraic Combinatoric, Hagenberg, Austria, 2009, pp. 493-504.

<code perl>
> print $C->N_VERTICES, " ", $C->DIM;
> print rows_labeled($C->VERTICES_IN_FACETS);
</code>
There are two disjoint facets covering all the vertices. Beware the numbering of facets depends on the convex hull algorithm employed.

<code perl>
> print $C->VERTICES_IN_FACETS->[8];
> print $C->VERTICES_IN_FACETS->[22];
> print rows_labeled($M);
</code>
Here is another polytope which is somewhat similar but not quite the same.

<code perl>
> $cross5=cross(5);
> print isomorphic($C,$cross5);
> print isomorphic($C->GRAPH->ADJACENCY,$cross5->GRAPH->ADJACENCY);
> print $cross5->F_VECTOR - $C->F_VECTOR;
</code>
Look at two facets of the five-dimensional cross polytope and their positions in the dual graph.

<code perl>
> print $cross5->VERTICES_IN_FACETS->[12];
> print $cross5->VERTICES_IN_FACETS->[13];
> print rows_labeled($cross5->DUAL_GRAPH->ADJACENCY);
</code>
Now we construct a new graph by manipulating the dual graph of the cross polytope by contracting a perfect matching.

<code perl>
> $g=new props::Graph($cross5->DUAL_GRAPH->ADJACENCY);
> $g->contract_edge(12,13);
> $g->contract_edge(24,26);
> $g->contract_edge(17,21);
> $g->contract_edge(3,11);
> $g->contract_edge(6,22);
> $g->squeeze;
</code>
The last command renumbers the nodes sequentially, starting from 0. This is necessary to render the graph a valid object.

<code perl>
> print isomorphic($C->DUAL_GRAPH->ADJACENCY,$g);
</code>
This finally reveals the combinatorial structure: The cone //C// is a cone over a 5-polytope which can be obtained from the 5-dimensional cross polytope by `''straightening'' five pairs of adjacent (simplex) facets into bipyramids over 3-simplices.