New Bounds for the Traveling Salesman Constant

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 $β$ such that $$ \lim_{n \rightarrow \infty}{n^{-1/2}L(X_1, \dots, X_n)} = β\qquad \mbox{almost surely.}$$ The best bounds for $β$ are still the ones originally established by Beardwood, Halton $\&$ Hammersley $0.625 \leq β\leq 0.922$. We slightly improve both upper and lower bounds.

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