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Greg Egan
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Addressing the moments in dimensions greater than 2, if $A_d$ is the area of a triangle whose vertices are chosen uniformly at random from a unit sphere in $\mathbb{R}^d$, the even moments I’ve computed with symbolic integration for $d=3$ all satisfy:

$$E(A_3^{2m})=\frac{3 (4 m+2) (m!)^3 (6 m+1)!}{(3 m)! (2 m+1)! (4 m+3)!}$$

I’ve checked this up to $m=15$. The first 10 values are:

$$\frac{1}{2},\frac{13}{30},\frac{323}{700},\frac{23}{42},\frac{4495}{6468},\frac{33263}{36036},\frac{65231}{51480},\frac{21460999}{12033450},\frac{30741431}{12009140},\frac{8965109}{2401828}$$

Greg Egan
  • 2.9k
  • 1
  • 16
  • 22