On the steiner ratio in \mathcal r- n

The Steiner minimum tree and the minimum spanning tree are two important problems in combinatorial optimization. Let P denote a finite set of points, called terminals, in the Euclidean space. A Steiner minimum tree of P, denoted by SMT(P), is a network with minimum length to interconnect all terminals, and a minimum spanning tree of P, denoted by MST(P), is also a minimum network interconnecting all the points in P, however, subject to the constraint that all the line segments in it have to terminate at terminals. Therefore, SMT(P) may contain points not in P, but MST(P) cannot contain such kind of points. Let R - n denote the n-dimensional Euclidean space. The Steiner ratio in R - n is defined to be rho(R - n ) = inf{L- s (P) \L- m (P) :P R - n , where L_s(P) and L_m(P), respectively, denote lengths of a Steiner minimum tree and a minimum spanning tree of P. The best previously known lower bound for \rho(R - n ) in the literature is 0.615. In this paper, we show that rho(R - n )>0.62 for any n ≥ 2.