On the Geometry of the Birkhoff Polytope. II. The Schatten $p$-norms

Abstract: In the first of this series of two articles, we studied some geometrical aspects of the Birkhoff polytope, the compact convex set of all $n \times n$ doubly stochastic matrices, namely the Chebyshev center, and the Chebyshev radius of the Birkhoff polytope associated with metrics induced by the operator norms from $\ell_n^p$ to $\ell_n^p$ for $1 \leq p \leq \infty$. 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 $1 \leq p < \infty$. While studying these properties, the intrinsic connection to the minimal trace, which naturally appears in the assignment problem, is also established.