An Identity Relating Moments of Functionalsof Convex Hulls

Denote by $K_n$ the convex hull of $n$ independent random pointsdistributed uniformly in a convex body $K$ in $R^d$, by $V_n$ the volume of$K_n$, by $D_n$ the volume of $Kbackslash K_n$, and by $N_n$ the number ofvertices of $K_n$. A well-known identity due to Efron relates the expectedvolume ${it ED}_n$---and thus ${it EV}_n$---to the expected number ${it EN}_{n+1}$. Thisidentity is extended from expected values to higher moments.The planar case of the arising identity for the variances provides in a simpleway the corrected version of a central limit theorem for $D_n$ by Cabo andGroeneboom ($K$ being a convex polygon) and an improvement of a central limittheorem for $D_n$ by Hsing ($K$ being a circular disk). Estimates of $var D_n$($K$ being a two-dimensional smooth convex body) and $var N_n$ ($K$ being a$d$-dimensional smooth convex body, $dgeq 4$) are obtained.The identity for moments of arbitrary order shows that the distribution of $N_n$determines ${it EV}_{n-1}, {it EV}_{n-2}^2,dots, {it EV}_{d+1}^{n-d-1}$. Reversely it isproved that these $n-d-1$ moments determine the distribution of $N_n$ entirely.The resulting formula for the probability that $N_n=k (k=d+1,dots , n)$appears to be new for $kgeq d+2$ and yields an answer to a question raised byBaryshnikov. For $k=d+1$ the formula reduces to an identity which has beenrepeatedly pointed out.