Singularities of Minimal Surfaces and Networks and Related Extremal Problems in Minkowski Space.

This paper describes results on two questions about points in a Minkowski space that arose in the study of minimal surfaces and networks with singularities. Let PHI denote a norm on R sup n having unit ball B . The first question concerns the maximal number of vectors in a PHI-equilateral set both for general norms and for strictly convex norms. A new proof is given of the known result that a PHI-equilateral set has cardinality at most 2 sup n for a general norm. There exists a strictly convex norm having a PHI-equilateral set of cardinality at least (1.02) sup n, for large n. The second question concerns the maximal number of PHI-unit vectors such that PHI (x sub i + x sub j) = 1 whenever i != j, both with and without the side condition SIGMA x sub i = 0. Here exponentially large sets exist without the side condition; with it there are at most 2n vectors in the set.