Let $P_m$ be a subset for R^mxm be the polytope given by:
$x_i,_j \ge 0$
$x_i,_1 + ... + x_i,_m \le 1$
$x_1,_j + ... + x_m,_j \le 1$
$\sum_{1 \le i,j \le m } \ x_i,_j \ge m-1$
Contruct a natural affine map $f$ from $P_n$ to the birkoff polytope B_n+1. Use $f$ to prove that the dimension of the birkoff polytope is
dim$B_n=(n-1)^2$