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Random approximation of convex sets *
Author(s) -
Schneider Rolf
Publication year - 1988
Publication title -
journal of microscopy
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.569
H-Index - 111
eISSN - 1365-2818
pISSN - 0022-2720
DOI - 10.1111/j.1365-2818.1988.tb04682.x
Subject(s) - mathematics , convex hull , convex body , combinatorics , regular polygon , independent and identically distributed random variables , convex set , convex analysis , convex combination , boundary (topology) , proper convex function , hausdorff space , sequence (biology) , distribution (mathematics) , mathematical analysis , geometry , random variable , convex optimization , statistics , biology , genetics
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.