## The Annals of Applied Probability

### Rates of Convergence of Means for Distance-Minimizing Subadditive Euclidean Functionals

Kenneth S. Alexander

#### Abstract

Functionals $L$ on finite subsets $A$ of $\mathbb{R}^d$ are considered for which the value is the minimum total edge length among a class of graphs with vertex set equal to, or in some cases containing, $A$. Examples include minimal spanning trees, the traveling salesman problem, minimal matching and Steiner trees. Beardwood, Halton and Hammersley, and later Steele, have shown essentially that for $\{X_1, \ldots, X_n\}$ a uniform i.i.d. sample from $\lbrack 0,1 \rbrack^d, EL(\{X_1, \ldots, X_n\})/n^{(d-1)/d}$ converges to a finite constant. Here we bound the rate of this convergence, proving a conjecture of Beardwood, Halton and Hammersley.

#### Article information

Source
Ann. Appl. Probab., Volume 4, Number 3 (1994), 902-922.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aoap/1177004976

Digital Object Identifier
doi:10.1214/aoap/1177004976

Mathematical Reviews number (MathSciNet)
MR1284990

Zentralblatt MATH identifier
0809.60016

JSTOR