An improved lower bound for the Traveling Salesman constant

Let X1,X2,…,Xn be independent uniform random variables on [0,1]2. Let L(X1,…,Xn) be the length of the shortest Traveling Salesman tour through these points. Beardwood et al (1959) showed that there exists a constant β such that limn→∞L(X1,…,Xn)n=βalmost surely. It was shown that β≥0.625. Building upon an approach proposed by Steinerberger (2015), we improve the lower bound to β≥0.6277.