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<p>In this paper, we extend two celebrated inequalities by Busemann—the random simplex inequality and the intersection inequality—to both complex and quaternionic vector spaces. Our proof leverages a monotonicity property under symmetrization with respect to complex or quaternionic hyperplanes. Notably, we demonstrate that the standard Steiner symmetrization, contrary to assertions in a paper by Grinberg, does not exhibit this monotonicity property.</p>

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Complex and quaternionic analogues of Busemann’s random simplex and intersection inequalities

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月4日

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  • Christos Saroglou,
  • Thomas Wannerer

摘要

In this paper, we extend two celebrated inequalities by Busemann—the random simplex inequality and the intersection inequality—to both complex and quaternionic vector spaces. Our proof leverages a monotonicity property under symmetrization with respect to complex or quaternionic hyperplanes. Notably, we demonstrate that the standard Steiner symmetrization, contrary to assertions in a paper by Grinberg, does not exhibit this monotonicity property.