  Let $X_1, X_2, \dots, X_n$ be independent and uniformly distributed random
variables in the unit square $[0,1]^2$ and let $L(X_1, \dots, X_n)$ be the
length of the shortest traveling salesman path through these points. In 1959,
Beardwood, Halton $\&$ Hammersley proved the existence of a universal constant
$\beta$ such that $$ \lim_{n \rightarrow \infty}{n^{-1/2}L(X_1, \dots, X_n)} =
\beta \qquad \mbox{almost surely.}$$ The best bounds for $\beta$ are still the
ones originally established by Beardwood, Halton $\&$ Hammersley $0.625 \leq
\beta \leq 0.922$. We slightly improve both upper and lower bounds.
