Precise asymptotic formulae are obtained for the expected number of k-faces of the orthogonal projection of a regular n-simplex in n-space onto a randomly chosen isotropic subspace of fixed dimension or codimension, as the dimension n tends to infinity.

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KEY WORDS. Crofton's theorem on mean values, expected volume of a random polytope, geometric probabilities, inscribed random polytopes, integral geometry, set of uniform random points, stochastic geometry, Sylvester's problem. SUMMARYFor any convex body K in d-dimensional Euclidean space Ed(d32) and for integers n and i, n3d+ 1, l s i s n , let Vf& ,(K) be the expected volume of the convex hull Hm-:, 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 Y , , ;(K) for the case that Kis a d-dimensional unit ball by considering an adequate decomposition of Hfi-:, I into d-dimensional simplices.To solve the important case i=O, 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.

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APROXIMACIÓN ALEATORIA DE CUERPOS CONVEXOS* FERNANDO AFFENTRANGERProblems related te the random approximation of convex bodies fall into the field of integral geometry and geometric probabilities . The aim of this paper is te give a survey of known results about the stochastic model that has received special attention in the literature and that can be described as follows :Let K be a d-dimensional convex body in Euclidean space Rd, d >_ 2 . Denote by H the convex hull of n independent random points X 1, . . . , X distributed identically and uniformly in the interior of K. lf cp is a random variable en d-dimensional polytopes in Rd, we define the random variable cp, by ,pn = W(conv{X1, . . . ,Xn}), *Extended version of an invited talk,