On Steiner minimal trees withLp distance

LetL_p be the plane with the distanced_p(A₁,A₂) = (¦x₁ –x₂¦^p + ¦y₁ –y₂¦^p)^/1p wherex_i andy_i are the cartesian coordinates of the pointA_i. LetP be a finite set of points inL_p. We consider Steiner minimal trees onP. It is proved that, for 1 <p < , each Steiner point is of degree exactly three. Define the Steiner ratio _p to be inf{L_s(P)/L_m(P)¦PL_p} whereL_s(P) andL_m(P) are lengths of the Steiner minimal tree and the minimal spanning tree onP, respectively. Hwang showed ₁ = 2/3. Chung and Graham proved ₂ > 0.842. We prove in this paper that {} = 2/3 and (2/2)_{12 p} 3/2 for anyp.This work was supported in part by the National Science Foundation of China and the President Foundation of Academia Sinica.