167 citations
- Chemnitz University of TechnologyDE1 paper
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7 papers · 1 filter
On the existence of shortest directed networks
Konrad J Swanepoel
A directed network connecting a set A to a set B is a digraph containing an a-b path for each a in A and b in B. Vertices in the directed network not in A or B are called Steiner p…
Balancing unit vectors
Konrad J. Swanepoel
Theorem A. Let be unit vectors in a normed plane. Then there exist signs $\epsi_1,...,\epsi_{2k+1}\in\{\pm 1\}$ such that $\norm{\sum_{i=1}^{2k+1}\epsi_i x_i}\le…
Vertex degrees of Steiner Minimal Trees in and other smooth Minkowski spaces
K. J. Swanepoel
We find upper bounds for the degrees of vertices and Steiner points in Steiner Minimal Trees in the d-dimensional Banach spaces \ell_p^d independent of d. This is in contrast to Mi…
Cardinalities of k-distance sets in Minkowski spaces
Konrad J. Swanepoel
A subset of a metric space is a k-distance set if there are exactly k non-zero distances occuring between points. We conjecture that a k-distance set in a d-dimensional Banach spac…
The geometry of Minkowski spaces -- a survey. Part I
Horst Martini, Konrad J Swanepoel, Gunter Weiss
We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place sp…
Extremal Problems in Minkowski Space related to Minimal Networks
Konrad J Swanepoel
We solve the following problem of Z. Füredi, J. C. Lagarias and F. Morgan [FLM]: Is there an upper bound polynomial in for the largest cardinality of a set S of unit vectors in…