An approach for proving lower bounds: solution of Gilbert-Pollak's conjecture on Steiner ratio

A family of finitely many continuous functions on a polytope X, namely (g/sub i/(x))/sub i in I/, is considered, and the problem of minimizing the function f(x)=max/sub i in I/g/sub i/(x) on X is treated. It is shown that if every g/sub i/(x) is a concave function, then the minimum value of f(x) is achieved at finitely many special points in X. As an application, a long-standing problem about Steiner minimum trees and minimum spanning trees is solved. In particular, if P is a set of n points on the Euclidean plane and L/sub s/(P) and L/sub m/(P) denote the lengths of a Steiner minimum tree and a minimum spanning tree on P, respectively, it is proved that, for any P, L/sub S/(P)>or= square root 3L/sub m/(P)/2, as conjectured by E.N. Gilbert and H.O. Pollak (1968).<>