ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

13. Combinatorics

1991 MS Classification: 53,58,52 Ivanov, Alexandr, Moscow State University, Russia Linear Networks in Euclidean Spaces R^{n} Let [G, φ] = R^{ns} be the configuration space of all movable vertices of linear networks with fixed topology G and given boundary φ. Let [G, φ]_{min} be the set of all minima points of the length functional on [G, φ]. Consider the set M_{G}(φ) of all local minimal networks of topology G with the boundary φ. F. Hwang proved that for a tree G the set M_{G}(φ) is at most pointwize. We generalize Hwang’s theorem for any topology G and any dimension n in terms of the structure of the set M_{G}(φ). In particular, we prove that M_{G}(φ) is a convex polyhedron, and we caculate its dimension. Refs. : A. O. Ivanov, A. A. Tuzhilin, Geometry of the set of minimal networks with given boundary and topology. Izvestia RAN, ser. matem., 1997, v.61, pp.119–152.