## The Annals of Applied Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Secondary: 05C80: Random graphs [See also 60B20] 90C27: Combinatorial optimization

**Keywords**

Subadditive Euclidean functional traveling salesman problem minimal spanning tree Steiner tree minimal matching

#### Citation

Alexander, Kenneth S. Rates of Convergence of Means for Distance-Minimizing Subadditive Euclidean Functionals. Ann. Appl. Probab. 4 (1994), no. 3, 902--922. doi:10.1214/aoap/1177004976. https://projecteuclid.org/euclid.aoap/1177004976