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Let \(\textbf{a}\) and \(\textbf{b}\) be vectors in \(\textbf{R}^n\) with nonnegative coordinates. Permuting the coordinates, one can assume that \(a_1 \ge \cdots \ge a_n\) and \(b_1 \ge \cdots \ge b_n\) . The vector \(\textbf{a}\) majorizes the vector \(\textbf{b}\) if \(\sum _{i=1}^k b_i \le \sum _{i=1}^k a_i\) for all \(k \in \{1,\ldots ,n-1\}\) and \(\sum _{i=1}^n b_i = \sum _{i=1}^n a_i\) . This paper exposes theorems of Hardy-Littlewood-Pólya and of Rado: The following are equivalent: (1) The vector \(\textbf{a}\) majorizes the vector \(\textbf{b}\) , (2) \(P\textbf{a} = \textbf{b}\) for some doubly stochastic matrix P, and (3) \( \textbf{b}\) is in the \(S_n\) -permutohedron generated by \(\textbf{a}\) .

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The Muirhead-Rado Inequality, 1: Vector Majorization and the Permutohedron

  • 会议论文

  • 发表日期：

    2025年

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  • Melvyn B. Nathanson

摘要

Let \(\textbf{a}\) and \(\textbf{b}\) be vectors in \(\textbf{R}^n\) with nonnegative coordinates. Permuting the coordinates, one can assume that \(a_1 \ge \cdots \ge a_n\) and \(b_1 \ge \cdots \ge b_n\) . The vector \(\textbf{a}\) majorizes the vector \(\textbf{b}\) if \(\sum _{i=1}^k b_i \le \sum _{i=1}^k a_i\) for all \(k \in \{1,\ldots ,n-1\}\) and \(\sum _{i=1}^n b_i = \sum _{i=1}^n a_i\) . This paper exposes theorems of Hardy-Littlewood-Pólya and of Rado: The following are equivalent: (1) The vector \(\textbf{a}\) majorizes the vector \(\textbf{b}\) , (2) \(P\textbf{a} = \textbf{b}\) for some doubly stochastic matrix P, and (3) \( \textbf{b}\) is in the \(S_n\) -permutohedron generated by \(\textbf{a}\) .