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February 1999 Asymptotics for the length of a minimal triangulation on a random sample
J. E. Yukich
Ann. Appl. Probab. 9(1): 27-45 (February 1999). DOI: 10.1214/aoap/1029962596


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.


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J. E. Yukich. "Asymptotics for the length of a minimal triangulation on a random sample." Ann. Appl. Probab. 9 (1) 27 - 45, February 1999.


Published: February 1999
First available in Project Euclid: 21 August 2002

zbMATH: 0931.68046
MathSciNet: MR1682604
Digital Object Identifier: 10.1214/aoap/1029962596

Primary: 60D05 , 60F15
Secondary: 68C05 , 68E10

Keywords: discrete probabilistic Plateau problem , subadditive and superadditive Euclidean functionals , Traveling salesman problem , Triangulation

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.9 • No. 1 • February 1999
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