On the Geometry of the Birkhoff Polytope. I. The operator ${\mathrm{\ell }}_n^p$-norms

The geometry of the Birkhoff polytope, i.e., the compact convex set of all $n\times n$ doubly stochastic matrices, has been an active subject of research. While its faces, edges and facets as well as its volume have been intensely studied, other geometric characteristics such as the center and radius were left off, despite their natural uses in some areas of mathematics. In this paper, we completely characterize the Chebyshev center and the Chebyshev radius of the Birkhoff polytope associated with the metrics induced by the operator ${\mathrm{\ell }}_n^p$-norms for the range $1\le p\le \mathrm{\infty }$.