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In Theorem 6.13 , we stated formulas for the moments of the volume of random beta-type simplices generated by k random points in \(\mathbb {R}^d\) . In this chapter, we present alternative methods for deriving these formulas. For simplices with \(k=2\) vertices—that is, for segments—we derive moment formulas using properties of Bessel functions. For simplices with \(k\geq 3\) vertices, we present an inductive geometric argument that relies on the base-times-height formula for the volume of a simplex.

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Alternative Approaches to Volumes of Beta-Type Simplices

  • 章节

  • 发表日期：

    2026年

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  • Zakhar Kabluchko,
  • David Albert Steigenberger,
  • Christoph Thäle

摘要

In Theorem 6.13 , we stated formulas for the moments of the volume of random beta-type simplices generated by k random points in \(\mathbb {R}^d\) . In this chapter, we present alternative methods for deriving these formulas. For simplices with \(k=2\) vertices—that is, for segments—we derive moment formulas using properties of Bessel functions. For simplices with \(k\geq 3\) vertices, we present an inductive geometric argument that relies on the base-times-height formula for the volume of a simplex.