The barycenter and minimum permanent of certain polytope of doubly stochastic matrices (Q2721576)

Assume that \(n\geq 3\). Let \(\Omega_n\) denote the set of all \(n\) by \(n\) doubly stochastic matrices, \(D_n:=[d_{ij}]\) be the \(n\) by \(n\) \((0, 1)\)-matrix satisfying that \(d_{ij}=0\) if and only if \(i=j=1\) or \(2\leq i, j\leq[\frac {n+1}2]\). Consider a face of the polytope \(\Omega_n\) associated with \(D_n\) of the type \(\Omega(D_n):=\{A=[a_{ij}]\in\Omega_n |a_{ij}=0\text{ if }d_{ij}=0\}\). NEWLINENEWLINENEWLINEThe authors identify the barycenter of \(\Omega(D_n)\), and show that the minimum value of the permanent function on \(\Omega(D_n)\) does not occur at its barycenter. Moreover, the minimizing matrix on \(\Omega(D_n)\) and its minimum permanent for \(n\) odd are determined. The following open problem is posed: determine the minimizing matrices and the minimum permanent on \(\Omega(D_n)\) for an even integer \(n\geq 6\).