The diameter of the Birkhoff polytope release_oaor5y77zjalnj5p5b5kzzpssa

Abstract

<jats:title>Abstract</jats:title> The geometry of the compact convex set of all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2023-0113_eq_001.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>×</m:mo> <m:mi>n</m:mi> </m:math> <jats:tex-math>n\times n</jats:tex-math> </jats:alternatives> </jats:inline-formula> 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 <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2023-0113_eq_002.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi>ℓ</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msubsup> </m:math> <jats:tex-math>{\ell }_{n}^{p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2023-0113_eq_003.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi>ℓ</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msubsup> </m:math> <jats:tex-math>{\ell }_{n}^{p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and the Schatten <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2023-0113_eq_004.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> <jats:tex-math>p</jats:tex-math> </jats:alternatives> </jats:inline-formula>-norms, both for the range <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2023-0113_eq_005.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mi>∞</m:mi> </m:math> <jats:tex-math>1\le p\le \infty </jats:tex-math> </jats:alternatives> </jats:inline-formula>, 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.
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