An explicit formula about the convex hull of random points (Q911137)

Denote by \(V_ n^{(d)}\) the expected volume of the convex hull of n points chosen independently according to a given probability measure \(\mu\) in Euclidean d-space \(E^ d\). For \(d=2,3\) and \(\mu\) the uniform distribution on a convex body in \(E^ d\), \textit{F. Affentranger} [ibid. 43, 39-45 and 151-152 (1988; see the preceding entries, Zbl 0697.60013 and Zbl 0697.60014)] has shown that \[ (1)\quad V^{(d)}_{d+2m}=\sum^{m}_{k=1}\gamma_ k\left( \begin{matrix} d+2m\\ 2k- 1\end{matrix} \right)V^{(d)}_{d+2m-2k+1}\quad (m=1,2,...), \] where the \(\gamma_ k\) can be obtained recursively from \[ \gamma_ 1=1/2,\quad 2\gamma_ k=1-\sum^{k-1}_{i=1}\left( \begin{matrix} 2k-1\\ 2i-1\end{matrix} \right)\gamma_ i\quad (k\geq 2). \] In this note we show that in formula (1) the \(\gamma_ k\) can be expressed explicitly by \[ \gamma_ k=(2^{2k}-1)B_{2k}/k\quad (k=1,2,...). \]