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Abstract. One of our results: Let X be a finite set on the plane, 0 < ε < 1. Then there exists a set F (a weak ε-net) of size at most 7/ε 2 such that every convex set containing at least ε|X| elements of X intersects F . Note that the size of F is independent of the size of X.

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ABSTRACT:A random spherical polytope P n in a spherically convex set K ⊂ S d as considered here is the spherical convex hull of n independent, uniformly distributed random points in K. The behaviour of P n for a spherically convex set K contained in an open halfsphere is quite similar to that of a similarly generated random convex polytope in a Euclidean space, but the case when K is a halfsphere is different. This is what we investigate here, establishing the asymptotic behaviour, as n tends to infinity, of the expectation of several characteristics of P n , such as facet and vertex number, volume and surface area. For the Hausdorff distance from the halfsphere, we obtain also some almost sure asymptotic estimates.

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