This tutorial is probably also available as a Jupyter notebook in the `demo`

folder in the polymake source and on github.

Different versions of this tutorial: latest release, release 3.6, release 3.5, release 3.4, nightly master

# 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 CarathÃ©odory's theorem. J. Reine Angew. Math. 510 (1999), 179-185.

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

> $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);

From

> 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

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*.

> $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

The following loop iterates over all invertible 6×6 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.

> foreach (@{all_subsets_of_k(range(0,9),6)}) { > $B = $M->minor($_,All); > if (det($B)) { > print lin_solve(transpose($B),$x), "\n"; > } > }

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 MÃ¼ller, 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.

> print $C->N_VERTICES, " ", $C->DIM; > print rows_labeled($C->VERTICES_IN_FACETS);

There are two disjoint facets covering all the vertices. Beware the numbering of facets depends on the convex hull algorithm employed.

> print $C->VERTICES_IN_FACETS->[8]; > print $C->VERTICES_IN_FACETS->[22]; > print rows_labeled($M);

Here is another polytope which is somewhat similar but not quite the same.

> $cross5=cross(5); > print isomorphic($C,$cross5); > print isomorphic($C->GRAPH->ADJACENCY,$cross5->GRAPH->ADJACENCY); > print $cross5->F_VECTOR - $C->F_VECTOR;

Look at two facets of the five-dimensional cross polytope and their positions in the dual graph.

> print $cross5->VERTICES_IN_FACETS->[12]; > print $cross5->VERTICES_IN_FACETS->[13]; > print rows_labeled($cross5->DUAL_GRAPH->ADJACENCY);

Now we construct a new graph by manipulating the dual graph of the cross polytope by contracting a perfect matching.

> $g=new props::Graph($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;

The last command renumbers the nodes sequentially, starting from 0. This is necessary to render the graph a valid object.

> print isomorphic($C->DUAL_GRAPH->ADJACENCY,$g);

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.