Denote by
KnK_n the convex hull of
nn independent random points
distributed uniformly in a convex body
KK in
\Rd\R^d, by
VnV_n the volume of
KnK_n, by
DnD_n the volume of
K\KnK\backslash K_n, and by
NnN_n the number of
vertices of
KnK_n. A well-known identity due to Efron relates the expected
volume
EDn{\it ED}_n---and thus
EVn{\it EV}_n---to the expected
number
ENn+1{\it EN}_{n+1}. This
identity is extended from expected values to higher moments.
The planar case of the arising identity for the variances provides in a simple
way the corrected version of a central limit theorem for
DnD_n by Cabo and
Groeneboom (
KK being a convex polygon) and an improvement of a central limit
theorem for
DnD_n by Hsing (
KK being a circular disk). Estimates of
\var Dn\var D_n
(
KK being a two-dimensional smooth convex body) and
\var Nn\var N_n (
KK being a
dd-dimensional smooth convex body,
d ³ 4d\geq 4) are obtained.
The identity for moments of arbitrary order shows that the distribution of
NnN_n
determines
EVn-1, EVn-22,..., EVd+1n-d-1{\it EV}_{n-1}, {\it EV}_{n-2}^2,\dots, {\it EV}_{d+1}^{n-d-1}. Reversely it is
proved that these
n-d-1n-d-1 moments determine the distribution of
NnN_n entirely.
The resulting formula for the probability that
Nn=k (k=d+1,... , n)N_n=k\ (k=d+1,\dots , n)
appears to be new for
k ³ d+2k\geq d+2 and yields an answer to a question raised by
Baryshnikov. For
k=d+1k=d+1 the formula reduces to an identity which has been
repeatedly pointed out.