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 \(1 \le p \le \infty \). 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 \(1 \le p < \infty \). While studying these properties, the intrinsic connection to the minimal trace, which naturally appears in the assignment problem, is also established.