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Asymptotics for the length of a minimal triangulation on a random sample. (English) Zbl 0931.68046

Summary: Given \(F\subset [0,1]^2\) and finite, let \(\sigma(F)\) denote the length of the minimal Steiner triangulation of points in \(F\). By showing that minimal Steiner triangulations fit into the theory of subadditive and superadditive Euclidean functionals, we prove under a mild regularity condition that \[ \lim_{n\to\infty} \sigma(X_1,\dots, X_n)/n^{1/2}= \beta \int_{[0,1]^2} f(x)^{1/2} dx\quad\text{c.c.}, \] where \(X_1,\dots, X_n\) are i.i.d. random variables with values in \([0,1]^2\), \(\beta\) is a positive constant, \(f\) is the density of the absolutely continuous part of the law of \(X_1\), and c.c. denotes complete convergence. This extends the work of Steele. The result extends naturally to dimension three and describes the asymptotics for the probabilistic Plateau functional, thus making progress on a question of Beardwood, Halton and Hammersley. Rates of convergence are also found.

MSC:

68Q05 Models of computation (Turing machines, etc.) (MSC2010)
68R10 Graph theory (including graph drawing) in computer science
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