Generalization of a formula of C. Buchta about the convex hull of random points (Q911134)

For an arbitrary convex body K in the d-dimensional Euclidean space \(E^ d\) (d\(\geq 2)\) we denote by \(V_ n^{(d)}(K)\) the expected volume of the convex hull \(H_ n\) of n random points chosen independently and uniformly inside K. For arbitrary plane convex sets, respectively three- dimensional convex bodies, \textit{C. Buchta} [ibid. 38, 153-156 (1983; Zbl 0521.52005)] proved the following relationships \[ V_ 4^{(2)}(K)=2V_ 3^{(2)}(K)\quad and\quad V_ 5^{(3)}(K)=(5/2)V_ 4^{(3)}(K). \] The aim of this note is to generalize Buchta's formulas.