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<p>An <i>n</i> × <i>m</i> array with nonnegative entries is called doubly stochastic if the sum of its entries at each row is <i>m</i> and at each column is <i>n</i>. The set of all <i>n</i> × <i>m</i> doubly stochastic arrays is a convex polytope with finitely many extremal points. The main result of this paper characterizes the possible sizes of the supports of all extremal <i>n</i> × <i>m</i> doubly stochastic arrays. In particular we prove that the minimal size of the support of an <i>n</i> × <i>m</i> doubly stochastic array is <i>n</i> + <i>m</i> − gcd(<i>n</i>, <i>m</i>). Moreover, for <i>m</i> = <i>kn</i> + 1 we also characterize the structure of the support of the extremal arrays.</p>

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Support of extremal doubly stochastic arrays

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年2月17日

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  • Mark Mordechai Etkind,
  • Nir Lev

摘要

An n × m array with nonnegative entries is called doubly stochastic if the sum of its entries at each row is m and at each column is n. The set of all n × m doubly stochastic arrays is a convex polytope with finitely many extremal points. The main result of this paper characterizes the possible sizes of the supports of all extremal n × m doubly stochastic arrays. In particular we prove that the minimal size of the support of an n × m doubly stochastic array is n + m − gcd(n, m). Moreover, for m = kn + 1 we also characterize the structure of the support of the extremal arrays.