On the Travelling Salesperson Problem in Many Dimensions

Consider d ⩾ 2, and m points X 1 , …, X m that are independent uniformly distributed in [0, 1] d . It is well known that the length T m of the shortest tour through X 1 , …, X m satisfies lim m → ∞ E ( T m )/ m 1−1/ d = β( d ) for a certain number β( d ). We show that for some numerical constant K ,\documentclass{article}\pagestyle{empty}\begin{document}$$ |\beta (d)\, - \,(d/2\pi e)^{1/2} | \le \,K(\log \,d)/d $$\end{document} .