THE STRUCTURE OF SINGULARITIES IN PHI-MINIMIZING NETWORKS IN R2

It is well known that length-minimizing networks in R2 consist of segments meeting only in threes. This paper considers uniformly convex norms-PHI more general than length. The first theorem says that for any such smooth-PHI, minimizing networks still meet only in threes. The second theorem shows that for some piecewise smooth-PHI, segments can meet in fours (although never in fives or more).