On the Delone Triangulation Numbers

“Almost exact” estimates for the Delone triangulation numbers are given. In particular lim n→∞ deln−1 n = 1 2πe = 0.0585498.... 1991 Mathematics Subject Classification. 52A22. The first named author was partially supported by the KBN grant no. 2 P03A 022 15. Both authors were partially supported by the Erwin-Schrödinger-Institute in Vienna. Typeset by AMS-TEX 1 2 PIOTR MANKIEWICZ CARSTEN SCHÜTT By B 2 we denote the Euclidean ball in R. Recall that the symmetric difference metric for convex bodies K and C is the volume of the symmeric difference of K and C dS(K,C) = voln(K4C). McClure and Vitale, [McVi], in dimension 2 and Gruber, [Gr], in arbitrary dimension obtained an asymptotic formula for convex bodies K in R with a C-boundary with everywhere positive curvature. Namely, for such bodies lim N→∞ N 2 n−1 inf{dS(K,PN )| PN ⊂ K, PN is a polytope with N vertices} is equal to 1 2 deln−1 (∫ ∂K κ(x) 1 n+1 dμ(x) ) n+1 n−1 , where μ is the surface measure, κ the Gauss -curvature, and deln−1 is a constant connected with the Delone triangulations. Thus deln−1 = lim N→∞ 2 inf{dS(K,PN )| PN ⊂ K, PN is a polytope with N vertices} (∫ ∂K κ(x) 1 n+1 dμ(x) ) n+1 n−1 N− 2 n−1 . In particular, for K = B 2 , we get deln−1 = lim N→∞ 2 inf{dS(B 2 , PN )| PN ⊂ B 2 , PN is a polytope with N vertices} (voln−1(∂B 2 )) n+1 n−1 N− 2 n−1 . It was shown in [GRS] that there are constants c1 and c2 such that c1n ≤ deln ≤ c2n . This result was refined in [MS]: n− 1 n+ 1 voln−1(B n−1 2 ) − 2 n−1 ≤ deln−1 ≤ 2voln−1(∂B 2 )− 2 n−1 . Let K be a convex body. We consider random polytopes with vertices (randomly) chosen from the boundary of the body K. The expected volume of such a random polytope is defined by