An Identity Relating Moments of Functionals of Convex Hulls

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 \K n , and by N n the number of vertices of K n . A well-known identity due to Efron relates the expected volume ED n -and thus EV n -to the expected number 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 varD n (K being a two-dimensional smooth convex body) and varN n (K being a d-dimensional smooth convex body, d ≥ 4) are obtained.The identity for moments of arbitrary order shows that the distribution of N n determines EV n−1 , EV 2 n−2 , . . . , EV n−d−1 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, . . . , n) appears to be new for k ≥ 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.

Extension of Deltheil's study on random points in a convex quadrilateral

“…Using Buchta's more recent theory [2] in combination with our results, we find that N 5 , the number of sides of the convex hull when n = 5, takes the values 3, 4, and 5 with probabilities 5 36 − ψ, 5 9 , and 11 36 + ψ, respectively. Here, ψ := 5ab 9(1 + a) 2 …”

Central limit theorems for random polytopes

“…For more information on random polytopes we refer to a recent survey article by Schneider [26], and for a comparison of random polytopes and best approximating polytopes to a survey article by Gruber [14]. In particular, if the underlying convex set K itself is a polytope, we mention the work of Bárány and Buchta [7] dealing with the expectation of f i (P n ), and Groeneboom [13] and Cabo and Groeneboom [10] who obtained in the planar case central limit theorems for f 0 ( n ) (but the stated asymptotic value for the variance in [10] appears to be incorrect, see Hüsler [18], page 111, and for a corrected version Buchta [9] and Finch and Hueter [12]). To the best of our knowledge the only central limit theorem holding in arbitrary dimensions is due to Hueter [17] for f 0 (P n ) where the random points are chosen with respect to the d-dimensional normal distribution.…”

Convex hulls of random walks: Large-deviation properties

“…For this case some mathematical results are known [37][38][39]. In particular the remaining areã A n = 1 − A(n) outside the convex hull is considered.…”