Discrete & Computational Geometry

, Volume 33, Issue 1, pp 125–142

# An Identity Relating Moments of Functionals of Convex Hulls

Article

DOI: 10.1007/s00454-004-1109-3

Buchta, C. Discrete Comput Geom (2005) 33: 125. doi:10.1007/s00454-004-1109-3
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## Abstract

Denote by $K_n$ the convex hull of $n$ independent random points distributed uniformly in a convex body $K$ in $\R^d$, by $V_n$ the volume of $K_n$, by $D_n$ the volume of $K\backslash K_n$, and by $N_n$ the number of vertices of $K_n$. A well-known identity due to Efron relates the expected volume ${\it ED}_n$---and thus ${\it EV}_n$---to the expected number ${\it EN}_{n+1}$. 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 $D_n$ by Cabo and Groeneboom ($K$ being a convex polygon) and an improvement of a central limit theorem 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, $d\geq 4$) 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 is proved 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 $k\geq d+2$ and yields an answer to a question raised by Baryshnikov. For $k=d+1$ the formula reduces to an identity which has been repeatedly pointed out.