user_guide:tutorials:latest:caratheodory

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 — user_guide:tutorials:latest:caratheodory [2020/01/22 09:02] (current) Line 1: Line 1: + ===== 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//: + + + > $M = new Matrix<​Rational>​([[0,​1,​0,​0,​0,​0],​ + > [0,​0,​1,​0,​0,​0],​ + > [0,​0,​0,​1,​0,​0],​ + > [0,​0,​0,​0,​1,​0],​ + > [0,​0,​0,​0,​0,​1],​ + > [1,​0,​2,​1,​1,​2],​ + > [1,​2,​0,​2,​1,​1],​ + > [1,​1,​2,​0,​2,​1],​ + > [1,​1,​1,​2,​0,​2],​ + > [1,​2,​1,​1,​2,​0]]);​ + >$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 * $x; + 8 15 19/2 19/2 17 13 17 13 9 13 13 17 8 19/2 13 17 15 19/2 15 15 19/2 17 11 15 8 8 8 + + ​ + The following loop iterates over all invertible 6x6 submatrices of //M// and computes the unique representation of //x// as a linear combination of the rows of the submatrix. The output (suppressed as it is too long) shows that each such linear combination requires at least one negative or one non-integral coefficient. + + + > foreach (@{all_subsets_of_k(range(0,​9),​6)}) { + >$B = $M->​minor($_,​All);​ + >   if (det($B)) { + > print lin_solve(transpose($B),​$x),​ "​\n";​ + > } + > } + ​ + This means that //x// cannot be represented as a non-negative linear combination of any six of the given generators of //C//. + + ==== Analyzing the combinatorics ==== + + The following is taken from + + * Michael Joswig, Benjamin Müller, and Andreas Paffenholz: ''​polymake''​ and lattice polytopes. In Christian Krattenthaler,​ Volker Strehl and Manuel Kauers (eds.), Proceedings of the 21th International Conference on Formal Power Series and Algebraic Combinatoric,​ Hagenberg, Austria, 2009, pp. 493-504. + + + > print$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 $C->​VERTICES_IN_FACETS->​;​ + > print$C->​VERTICES_IN_FACETS->​;​ + > 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 $cross5->​VERTICES_IN_FACETS->​;​ + > print$cross5->​VERTICES_IN_FACETS->​;​ + > 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. +
• user_guide/tutorials/latest/caratheodory.txt