The expected volume of a random polytope in a ball *

For any convex body K in d‐dimensional Euclidean space Ed(d≥2) and for integers n and i, n ≥ d + 1,1 ≤ i ≤ n, let V(d)n‐ii(K) be the expected volume of the convex hull Hn‐i, i of n independent random points, of which n‐i are uniformly distributed in the interior, the other i on the boundary of K. We develop an integral formula for V(d)n‐i, i(K) for the case that K is a d‐dimensional unit ball by considering an adequate decomposition of V(d)n‐i, i into d‐dimensional simplices. To solve the important case i = 0, that is the case in which all points are chosen at random from the interior of Bd, we require in addition Crofton's theorem on mean values. We illustrate the usefulness of our results by treating some special cases and by giving numerical values for the planar and the three‐dimensional cases.

[1]  C. Buchta,et al.  Stochastische Approximation konvexer Polygone , 1984 .

[2]  C. Buchta,et al.  Zufallspolygone in konvexen Vielecken. , 1984 .

[3]  H. Groemer,et al.  On the mean value of the volume of a random polytope in a convex set , 1974 .

[4]  Rolf Schneider,et al.  Random approximation of convex sets * , 1988 .

[5]  A. Rényi,et al.  über die konvexe Hülle von n zufällig gewählten Punkten , 1963 .

[6]  B. Efron.  The convex hull of a random set of points , 1965 .

[7]  Adrian Baddeley,et al.  Integrals on a moving manifold and geometrical probability , 1977, Advances in Applied Probability.

[8]  Christian Buchta,et al.  A note on the volume of a random polytope in a tetrahedron , 1986 .

[9]  T. Cover,et al.  Geometrical Probability and Random Points on a Hypersphere , 1967 .

[10]  P. Gruber.  Approximation of convex bodies , 1983 .

[11]  R. Tichy,et al.  Stochastical approximation of convex bodies , 1985 .

[12]  H. Groemer,et al.  On some mean values associated with a randomly selected simplex in a convex set. , 1973 .

[13]  Christian Buchta.  On a conjecture of R. E. Miles about the convex hull of random points , 1986 .

[14]  J. F. C. Kingman,et al.  Random secants of a convex body , 1969, Journal of Applied Probability.

[15]  C. Buchta,et al.  Random polytopes in a ball , 1984 .

[16]  R. Tichy,et al.  Random polytopes on the torus , 1985 .

[17]  L. Santaló.  Integral geometry and geometric probability , 1976 .

[18]  R. E. Miles.  ISOTROPIC RANDOM SIMPLICES , 1971 .