The Steiner ratio conjecture of Gilbert and Pollak is true.

@article{Du1990TheSR,
  title={The Steiner ratio conjecture of Gilbert and Pollak is true.},
  author={D. Z. Du and Frank K. Hwang},
  journal={Proceedings of the National Academy of Sciences of the United States of America},
  year={1990},
  volume={87 23},
  pages={9464-6}
}
Let P be a set of n points on the euclidean plane. Let Ls(P) and Lm(P) denote the lengths of the Steiner minimum tree and the minimum spanning tree on P, respectively. In 1968, Gilbert and Pollak conjectured that for any P, Ls(P) >/= (radical3/2)Lm(P). We provide an abridged proof for their conjecture in this paper.