Random Polytopes on the Torus

The expected volume of the convex hull of n random points chosen independently and uniformly on the d-dimensional torus is determined. In the 1860's J. J. Sylvester raised the problem of determining the expected area V$(C) of the convex hull of three points chosen independently and uniformly at random from a given plane convex body C of area one. For some special plane convex bodies the problem was solved by Woolhouse [16], Crofton [8] and Deltheil [9], e.g. ^(triangle) = 1/12, V3(parallelogram) = 11/144, Irregular hexagon) = 289/3888, V3 (ellipse) = 35/487T2. More generally, Alikoski [1] proved that ,„. -, . . 9cos2w + 52cosu) + 44 / 27r (1) ^(regular m-gon) =-——„ . ■>— 1 w = 36m2 sin w m For any convex polygon C and an arbitrary number n of random points, the expected area Vn(C) of their convex hull was determined in a recent paper [5]. For example, (2) Vn (triangle) n+l^fc' K=l (3) Vn (parallelogram) 3(n n+l . k = l k 11 2k ) (n + l)2"+1 In 1917 Blaschke [3, 4] succeeded in proving that, among all plane convex bodies C of area one, Vs(C) attains its minimum if C is an ellipse. In 1974 Groemer [11, 12] generalized this statement to d-dimensional convex bodies C of volume one and an arbitrary number n of points. The value of this minimum is known for d + 1 points in a d-dimensional ellipsoid (Kingman [13]) and for an arbitrary number of points in dimensions d = 2 and d = 3 (cf. [6]). An obvious question is to ask for the expected volume of the convex hull of random points chosen on a compact metric manifold. A subset C of a compact metric manifold is called convex (cf. Bangert [2], Walter [14] ) if for any two points x, y G C all geodesic segments (i.e. all curves of minimal length on the manifold joining x and y) are completely contained in C. The convex hull of a set is the smallest convex set (on the manifold) containing it. In the case of the d-dimensional sphere S^ = {x G Ed+1: \\x\\ = 1} (Ed+1 denotes (d+ l)-dimensional Euclidean space), the metric is defined by the minimal Euclidean length of all curves on S^ joining two points. A convex set on S^ is either contained in a hemisphere or is the sphere itself. If n points are contained Received by the editors January 17, 1984 and, in revised form, March 26, 1984. 1980 Mathematics Subject Classification. Primary 52A22; Secondary 60D05. ©1985 American Mathematical Society 0002-9939/85 $1.00 + S.25 per page 312 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use C. BUCHTA AND R. F. TICHY 313 in some common hemisphere, their convex hull has the same property with probability one. As shown by Wendel [15], the probability p„ ' that n points chosen independently and uniformly at random from (the normalized measure space) S^ lie on some common hemisphere is given by {> Pn 2"-1 ¿-" \ k ' k=0 x 7 Further, Cover and Efron [7] showed for the expected volume Vn of the convex hull of n such points, on condition that all points lie on some common hemisphere, that