The Gilbert Arborescence Problem

We investigate the problem of designing a minimum cost flow network interconnecting n sources and a single sink, each with known locations in a normed space and with associated flow demands. The network may contain any finite number of additional unprescribed nodes from the space; these are known as the Steiner points. For concave increasing cost functions, a minimum cost network of this sort has a tree topology, and hence can be called a Minimum Gilbert Arborescence (MGA). We characterise the local topological structure of Steiner points in MGAs, showing, in particular, that for a wide range of metrics, and for some typical real-world cost-functions, the degree of each Steiner point is 3.Comment: 19 pages, 7 figures. arXiv admin note: text overlap with arXiv:0903.212

Sets of unit vectors with small subset sums

We say that a family of m {xi}Ιi ε[m]\} vectors in a Banach space X satisfies the k-collapsing condition if the sum of any k of them has norm at most 1. Let C(k, d) denote the maximum cardinality of a k-collapsing family of unit vectors in a d-dimensional Banach space, where the maximum is taken over all spaces of dimension d. Similarly, let CB(k, d) denote the maximum cardinality if we require in addition that the m vectors sum to 0. The case k = 2 was considered by Füredi, Lagarias and Morgan (1991). These conditions originate in a theorem of Lawlor and Morgan (1994) on geometric shortest networks in smooth finite-dimensional Banach spaces. We show that CB(k, d) = max {k + 1, 2d} for all k, d ≥ 2. The behaviour of C(k, d) is not as simple, and we derive various upper and lower bounds for various ranges of k and d. These include the exact values C(k, d) = max {k + 1, 2d} in certain cases. We use a variety of tools from graph theory, convexity and linear algebra in the proofs: in particular the Hajnal–Szemerédi Theorem, the Brunn– Minkowski inequality, and lower bounds for the rank of a perturbation of the identity matrix

Discrete Geometry

[no abstract available

The local Steiner problem in finite-dimensional normed spaces

We develop a general method for proving that certain star configurations in finite-dimensional normed spaces are Steiner minimal trees. This method generalizes the results of Lawlor and Morgan (1994) that could only be applied to differentiable norms. The generalization uses the subdifferential calculus from convex analysis. We apply this method to two special norms. The first norm, occurring in the work of Cieslik, has unit ball the polar of the difference body of the n-simplex (in dimension 3 this is the rhombic dodecahedron). We determine the maximum degree of a given point in a Steiner minimal tree in this norm. The proof makes essential use of extremal finite set theory. The second norm, occurring in the work of Conger (1989), is the sum of the ℓ1-norm and a small multiple of the ℓ2 norm. For the second norm we determine the maximum degree of a Steiner point