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--- tutorial:caratheodory [2010/08/10 10:39] – joswig
+++ tutorial:caratheodory [2014/01/03 15:45] – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
-===== A Counter-example to an integer analogue to Caratheodory's Theorem =====
+===== 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:
@@ Line 43: / Line 45: @@
 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>
+polytope > print $C->N_VERTICES, " ", $C->DIM;
+polytope > print rows_labeled($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>
+polytope > print $C->VERTICES_IN_FACETS->[8];
+polytope > print $C->VERTICES_IN_FACETS->[22];
+</code>
+<code>
+polytope > print rows_labeled($M);
+</code>
+Here is another polytope which is somewhat similar but not quite the same.
+<code>
+polytope > $cross5=cross(5);
+polytope > print isomorphic($C,$cross5);
+polytope > print isomorphic($C->GRAPH->ADJACENCY,$cross5->GRAPH->ADJACENCY);
+</code>
+<code>
+polytope > 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>
+polytope > print $cross5->VERTICES_IN_FACETS->[12];
+polytope > print $cross5->VERTICES_IN_FACETS->[13];
+polytope > print rows_labeled($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>
+polytope > $g=new props::Graph($cross5->DUAL_GRAPH->ADJACENCY);
+polytope > $g->contract_edge(12,13);
+polytope > $g->contract_edge(24,26);
+polytope > $g->contract_edge(17,21);
+polytope > $g->contract_edge(3,11);
+polytope > $g->contract_edge(6,22);
+polytope > $g->squeeze;
+</code>
+The last command renumbers the nodes sequentially, starting from 0.  This is necessary to render the graph a valid object.
+<code>
+polytope > print isomorphic($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.