Random approximation of convex sets*

SUMMARY We consider convex hulls of independent, identically distributed random points, the distribution of which depends in some way on a given convex body. In the first part, we study an i. i. d. sequence of points on the boundary of a smooth convex body K in the plane and assume that their distribution has a positive continuous density with respect to arc length. The rate of the almost sure convergence of the area, perimeter and Hausdorff distance from K of the successive convex hulls of the random points towards the corresponding values of K is investigated. Also random polygons generated by intersecting random supporting halfplanes of K are considered. The second part gives a survey of convex hulls of random points in d ‐dimensional convex bodies, with special emphasis on the asymptotic behaviour of the expectations of some geometrically interesting functionals.

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