On the geometry of the Birkhoff polytope II: the Schatten p-norms

In the first of this series of two articles, we studied some geometrical aspects of the Birkhoff polytope, the compact convex set of all n×ndoubly stochastic matrices, namely the Chebyshev center, and the Chebyshev radius of the Birkhoff polytope associated with metrics induced by the operator norms from ℓnpto ℓnpfor 1≤p≤∞. In the present paper, we take another look at those very questions, but for a different family of matrix norms, namely the Schatten p-norms, for 1≤p<∞. While studying these properties, the intrinsic connection to the minimal trace, which naturally appears in the assignment problem, is also established.