Geometry and the vertices of the Birkhoff polytope












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The Birkhoff polytope $P(n)$ is defined to be the points in $mathbb{R}^{n^2}$ which correspond naturally to $n times n$ doubly stochastic matrices. Is it possible to prove that the vertices of the polytope are equidistant from the center of the polytope which is the matrix with $frac{1}{n}$ in all entries without explicitly proving the permutation matrices are the vertices? Or equivalently, is it possible to prove it has exactly n! vertices without explicitly proving the permutation matrixes are the vertices?






combinatorics euclidean-geometry combinatorial-geometry







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    $begingroup$


    The Birkhoff polytope $P(n)$ is defined to be the points in $mathbb{R}^{n^2}$ which correspond naturally to $n times n$ doubly stochastic matrices. Is it possible to prove that the vertices of the polytope are equidistant from the center of the polytope which is the matrix with $frac{1}{n}$ in all entries without explicitly proving the permutation matrices are the vertices? Or equivalently, is it possible to prove it has exactly n! vertices without explicitly proving the permutation matrixes are the vertices?






    combinatorics euclidean-geometry combinatorial-geometry







    share|cite|improve this question









    $endgroup$















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      0








      0





      $begingroup$


      The Birkhoff polytope $P(n)$ is defined to be the points in $mathbb{R}^{n^2}$ which correspond naturally to $n times n$ doubly stochastic matrices. Is it possible to prove that the vertices of the polytope are equidistant from the center of the polytope which is the matrix with $frac{1}{n}$ in all entries without explicitly proving the permutation matrices are the vertices? Or equivalently, is it possible to prove it has exactly n! vertices without explicitly proving the permutation matrixes are the vertices?






      combinatorics euclidean-geometry combinatorial-geometry







      share|cite|improve this question









      $endgroup$




      The Birkhoff polytope $P(n)$ is defined to be the points in $mathbb{R}^{n^2}$ which correspond naturally to $n times n$ doubly stochastic matrices. Is it possible to prove that the vertices of the polytope are equidistant from the center of the polytope which is the matrix with $frac{1}{n}$ in all entries without explicitly proving the permutation matrices are the vertices? Or equivalently, is it possible to prove it has exactly n! vertices without explicitly proving the permutation matrixes are the vertices?






      combinatorics euclidean-geometry combinatorial-geometry




      combinatorics euclidean-geometry combinatorial-geometry






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 15 at 4:21









      EgoKillaEgoKilla

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