The Annals of Applied Probability

Asymptotics for the length of a minimal triangulation on a random sample

J. E. Yukich

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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 $$\lim_{n \to \infty} \sigma(X_1,\dots, X_n)/n^{1/2} = \beta \int_{[0, 1]^2}f(x)^{1/2} dx \c.c.,$$ 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.

Article information

Ann. Appl. Probab., Volume 9, Number 1 (1999), 27-45.

First available in Project Euclid: 21 August 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 68C05 68E10

Triangulation subadditive and superadditive Euclidean functionals traveling salesman problem discrete probabilistic Plateau problem


Yukich, J. E. Asymptotics for the length of a minimal triangulation on a random sample. Ann. Appl. Probab. 9 (1999), no. 1, 27--45. doi:10.1214/aoap/1029962596.

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