The convex hull of a random set of points

SUMMARY Various expectations concerning the convex hull of N independently and identically distributed random points in the plane or in space are evaluated. Integral expressions are given for the expected area, expected perimeter, expected probability content and expected number of sides. These integrals are shown to be particularly simple when the underlying distribution is normal or uniform over a disk or sphere. In two recent papers Renyi & Sulanki (1963, 1964) have given expressions for the expected area, perimeter, and number of vertices of the convex hull of N independently and identically selected random points in the plane. In these papers, limit theorems for asymptotically large N receive the greatest attention. Here the emphasis will be on the development of convenient formulae for fixed values of N. Some new results for random convex hulls in the plane are derived, such as the expected probability content, and also various expectations concerned with random convex hulls in three and higher dimensions. Special attention is given to the case of normally distributed points and also to the case of points drawn uniformly from an ellipse or an ellipsoid. Historically, calculating the expected probability content of three random points in two dimensions is known as 'Sylvester's problem', and has been solved explicitly for many different distributions by Deltheil (1926, p. 42). The corresponding problem of four points in three dimensions is connected with the name of Hostinsky (1925). A discussion of Sylvester's problem is given in Kendall & Moran's monograph, Geometrical Probability (1963). 2. THE EXPECTED NUMBER OF VERTICES, FACES, AND EDGES