Subadditive Euclidean Functionals and Nonlinear Growth in Geometric Probability

A limit theorem is established for a class of random processes (called here subadditive Euclidean functionals) which arise in problems of geometric probability. Particular examples include the length of shortest path through a random sample, the length of a rectilinear Steiner tree spanned by a sample, and the length of a minimal matching. Also, a uniform convergence theorem is proved which is needed in Karp's probabilistic algorithm for the traveling salesman problem. Disciplines Geometry and Topology | Other Mathematics Comments At the time of publication, author John M Steele was affiliated with the Stanford University. Currently ( June 2016), he is a faculty member in the Information and Decisions Department of the Wharton School at the University of Pennsylvania. This journal article is available at ScholarlyCommons: http://repository.upenn.edu/oid_papers/278 Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve, and extend access to The Annals of Probability. www.jstor.org ® The Annals of Proba~il ty 1981, Vol. 9, No.3, 365-376 SUBADDITIVE EUCLIDEAN FUNCTIONALS AND NONLINEAR GROWTH IN GEOMETRIC PROBABILITY BY J. MICHAEL STEELE Stanford University A limit theorem is established for a class of random processes (called here subadditive Euclidean functionals) which arise in problems of geometric probability. Particular examples include the length of shortest path through a random sample, the length of a rectilinear Steiner tree spanned by a sample, and the length of a minimal matching. Also, a uniform convergence theorem is proved which is needed in Karp's probabilistic algorithm for the traveling salesman problem.

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