Disproofs of Generalized Gilbert–Pollak Conjecture on the Steiner Ratio in Three or More Dimensions

AbstractThe Gilbert–Pollak conjecture, posed in 1968, was the most important conjecture in the area of “Steiner trees.” The “Steiner minimal tree” (SMT) of a point setPis the shortest network of “wires” which will suffice to “electrically” interconnectP. The “minimum spanning tree” (MST) is the shortest such network when onlyintersite line segmentsare permitted. The generalized GP conjecture stated thatρd=infP⊂Rd(lSMT(P)/lMST(P)) was achieved whenPwas the vertices of a regulard-simplex. It was showed previously that the conjecture is true ford=2 and false for 3⩽d⩽9. We settle remaining cases completely in this paper. Indeed, we show that any point set achievingρdmust have cardinality growing at least exponentially withd. The real question now is: What are the true minimal-ρpoint sets? This paper introduces the “d-dimensional sausage” point sets, which may have a lit to do with the answer

The Length of a Minimal Tree With a Given Topology: generalization of Maxwell Formula

The classic Maxwell formula calculates the length of a planar locally minimal binary tree in terms of coordinates of its boundary vertices and directions of incoming edges. However, if an extreme tree with a given topology and a boundary has degenerate edges, then the classic Maxwell formula cannot be applied directly, to calculate the length of the extreme tree in this case it is necessary to know which edges are degenerate. In this paper we generalize the Maxwell formula to arbitrary extreme trees in a Euclidean space of arbitrary dimension. Now to calculate the length of such a tree, there is no need to know either what edges are degenerate, or the directions of nondegenerate boundary edges. The answer is the maximum of some special linear function on the corresponding compact convex subset of the Euclidean space coinciding with the intersection of some cylinders.Comment: 6 ref

Non-crossing of plane minimal spanning and minimal T1 networks

AbstractFor any given collection of Euclidean plane points it will be shown that a minimal length T1 network (or 3-size quasi Steiner network (Du et al., 1991)) will intersect a minimal spanning tree only at the given Euclidean points