user_guide:tutorials:caratheodory

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 user_guide:tutorials:caratheodory [2019/01/25 09:27]oroehrig ↷ Page moved from tutorial:caratheodory to user_guide:caratheodory user_guide:tutorials:caratheodory [2019/02/11 23:09] (current) 2019/01/25 09:38 oroehrig ↷ Page moved from user_guide:caratheodory to user_guide:tutorials:caratheodory2019/01/25 09:27 oroehrig ↷ Page moved from tutorial:caratheodory to user_guide:caratheodory2017/06/13 11:27 oroehrig added some formatting to enable automated tests2014/01/03 15:45 external edit2010/08/10 10:59 joswig 2010/08/10 10:39 joswig 2010/08/10 10:37 joswig created 2019/01/25 09:38 oroehrig ↷ Page moved from user_guide:caratheodory to user_guide:tutorials:caratheodory2019/01/25 09:27 oroehrig ↷ Page moved from tutorial:caratheodory to user_guide:caratheodory2017/06/13 11:27 oroehrig added some formatting to enable automated tests2014/01/03 15:45 external edit2010/08/10 10:59 joswig 2010/08/10 10:39 joswig 2010/08/10 10:37 joswig created Line 1: Line 1: - ===== A Counter-example to an integer analog to Caratheodory'​s Theorem ===== + {{page>​.:​latest:​@FILEID@}} - ==== 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>​ - polytope > $M = new Matrix<​Rational>​([[0,​1,​0,​0,​0,​0],​ - polytope (2)> [0,​0,​1,​0,​0,​0],​ - polytope (3)> [0,​0,​0,​1,​0,​0],​ - polytope (4)> [0,​0,​0,​0,​1,​0],​ - polytope (5)> [0,​0,​0,​0,​0,​1],​ - polytope (6)> [1,​0,​2,​1,​1,​2],​ - polytope (7)> [1,​2,​0,​2,​1,​1],​ - polytope (8)> [1,​1,​2,​0,​2,​1],​ - polytope (9)> [1,​1,​1,​2,​0,​2],​ - polytope (10)> [1,​2,​1,​1,​2,​0]]);​ - polytope >$C=new Polytope<​Rational>​(POINTS=>​$M);​ - ​ - - From - <​code>​ - polytope > print$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//. - <​code>​ - polytope > $x=new Vector<​Rational>​([9,​13,​13,​13,​13,​13]);​ - polytope > print$C->​FACETS * $x; - 8 15 19/2 19/2 17 13 17 13 9 13 13 17 8 19/2 13 17 15 19/2 15 15 19/2 17 11 15 8 8 8 - ​ - - The following loop iterates over all invertible 6x6 submatrices of //M// and computes the unique representation of //x// as a linear combination of the rows of the submatrix. ​ The output (suppressed as it is too long) shows that each such linear combination requires at least one negative or one non-integral coefficient. - <​code>​ - > foreach (all_subsets_of_k(6,​0..9)) { - >$B = $M->​minor($_,​All);​ - >   if (det($B)) { - > print lin_solve(transpose($B),​$x),​ "​\n";​ - > } - > } - ​ - This means that //x// cannot be represented as a non-negative linear combination of any six of the given generators of //C//. - - ==== Analyzing the combinatorics ==== - - The following is taken from - * Michael Joswig, Benjamin Müller, and Andreas Paffenholz: ''​polymake''​ and lattice polytopes. ​ In Christian Krattenthaler,​ Volker Strehl and Manuel Kauers (eds.), Proceedings of the 21th International Conference on Formal Power Series and Algebraic Combinatoric,​ Hagenberg, Austria, 2009, pp. 493-504. - - <​code>​ - polytope > print$C->​N_VERTICES,​ " ", $C->DIM; - polytope > print rows_labeled($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. - <​code>​ - polytope > print $C->​VERTICES_IN_FACETS->​[8];​ - polytope > print$C->​VERTICES_IN_FACETS->​[22];​ - ​ - - <​code>​ - polytope > print rows_labeled($M);​ - ​ - - Here is another polytope which is somewhat similar but not quite the same. - <​code>​ - polytope >$cross5=cross(5);​ - polytope > print isomorphic($C,​$cross5);​ - polytope > print isomorphic($C->​GRAPH->​ADJACENCY,​$cross5->​GRAPH->​ADJACENCY);​ - ​ - - <​code>​ - polytope > print $cross5->​F_VECTOR -$C->​F_VECTOR;​ - ​ - 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($cross5->​DUAL_GRAPH->​ADJACENCY);​ - ​ - - Now we construct a new graph by manipulating the dual graph of the cross polytope by contracting a perfect matching. - <​code>​ - polytope >$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 > $g->​squeeze;​ - ​ - The last command renumbers the nodes sequentially,​ starting from 0. This is necessary to render the graph a valid object. - <​code>​ - polytope > print isomorphic($C->​DUAL_GRAPH->​ADJACENCY,​\$g);​ - ​ - 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/caratheodory.txt