A NEW BOUND FOR EUCLIDEAN STEINER MINIMAL TREES

Let X denote a finite set of points in the (Euclidean) plane. A minimum spanning tree for X , denoted by M ( X ) , is a connected network of Line segments joining various pairs of points of X so that the sum of the lengths of the line segments, called the length of M ( X ) , is as small as possible. We denote this minimum possible total length by LM(X). It may happen for a set X that by embedding X in a suitable larger set Y , L M ( Y ) is actually less than LM(X) . For example, if X consists of the three vertices of an equilateral triangle and Y consists of X together with the centroid of the triangle, then an easy calculation shows that