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arXiv · 2310.14043

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

Ludovick Bouthat·Javad Mashreghi·Frédéric Morneau-Guérin
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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.

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Ludovick Bouthat, Javad Mashreghi, Frédéric Morneau-Guérin. 2023-10-21. On the Geometry of the Birkhoff Polytope. II. The Schatten $p$-norms. https://doi.org/10.1007/s44146-024-00153-7

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