The diameter of the Birkhoff polytope

Abstract

The geometry of the compact convex set of all n × n n\times n doubly stochastic matrices, a structure frequently referred to as the Birkhoff polytope, has been an active subject of research as of late. Geometric characteristics such as the Chebyshev center and the Chebyshev radius with respect to the operator norms from ℓ n p {\ell }_{n}^{p} to ℓ n p {\ell }_{n}^{p} and the Schatten p p -norms, both for the range 1 ≤ p ≤ ∞ 1\le p\le \infty , have only recently been studied in depth. In this article, we continue in this vein by determining the diameter of the Birkhoff polytope with respect to the metrics induced by the aforementioned matrix norms.