The Exact Distribution of the Number of Vertices of a Random Convex Chain

Assume that n points P1, . . . , Pn are distributed independently and uniformly in the triangle with vertices (0, 1), (0, 0), and (1, 0). Consider the convex hull of (0, 1), P1, . . . , Pn, and (1, 0). The vertices of the convex hull form a convex chain. Let p (n) k be the probability that the convex chain consists — apart from the points (0, 1) and (1, 0) — of exactly k of the points P1, . . . , Pn. Bárány, Rote, Steiger, and Zhang [3] proved that p (n) n = 2/[n!(n+1)!]. We determine for k = 1, . . . , n−1 the values of p (n) k and thus obtain the distribution of the number of vertices of a random convex chain. Knowing this distribution provides the key to the answer of some long-standing questions in geometrical probability. Mathematics Subject Classification (2000) 52A22 60D05