The steiner ratio conjecture is true for five points

AbstractThe long-standing conjecture of Gilbert and Pollak states that for any set of n given points in the Euclidean plane, the ratio of the length of a Steiner minimal tree and the length of a minimal (spanning) tree is at least 32. This conjecture was shown to be true for n = 3 by Gilbert and Pollak, and for n = 4 by Pollak. Recently, Du, Yao and Hwang used a different approach to give a shorter proof for n = 4. In this paper we continue this approach to prove the conjecture for n = 5. Such results for small n are useful in obtaining bounds for the ratio of the two lengths in the general case