Article

Absorbing angles, Steiner minimal trees, and antipodality

Journal of Optimization Theory and Applications (Impact Factor: 1.42). 08/2011; 143(1). DOI: 10.1007/s10957-009-9552-1
Source: arXiv

ABSTRACT We give a new proof that a star $\{op_i:i=1,...,k\}$ in a normed plane is a
Steiner minimal tree of its vertices $\{o,p_1,...,p_k\}$ if and only if all
angles formed by the edges at o are absorbing [Swanepoel, Networks \textbf{36}
(2000), 104--113]. The proof is more conceptual and simpler than the original
one.
We also find a new sufficient condition for higher-dimensional normed spaces
to share this characterization. In particular, a star $\{op_i: i=1,...,k\}$ in
any CL-space is a Steiner minimal tree of its vertices $\{o,p_1,...,p_k\}$ if
and only if all angles are absorbing, which in turn holds if and only if all
distances between the normalizations $\frac{1}{\|p_i\|}p_i$ equal 2. CL-spaces
include the mixed $\ell_1$ and $\ell_\infty$ sum of finitely many copies of
$R^1$.

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    ABSTRACT: The subject of this monograph can be described as the local properties of geometric Steiner minimal trees in finite-dimensional normed spaces. A Steiner minimal tree of a finite set of points is a shortest connected set interconnecting the points. For a quick introduction to this topic and an overview of all the results presented in this work, see Chapter 1. The relevant mathematical background knowledge needed to understand the results and their proofs are collected in Chapter 2. In Chapter 3 we introduce the Fermat-Torricelli problem, which is that of finding a point that minimizes the sum of distances to a finite set of given points. We only develop that part of the theory of Fermat-Torricelli points that is needed in later chapters. Steiner minimal trees in finite-dimensional normed spaces are introduced in Chapter 4, where the local Steiner problem is given an exact formulation. In Chapter 5 we solve the local Steiner problem for all two-dimensional spaces, and generalize this solution to a certain class of higher-dimensional spaces (CL spaces). The twodimensional solution is then applied to many specific norms in Chapter 6. Chapter 7 contains an abstract solution valid in any dimension, based on the subdifferential calculus. This solution is applied to two specific high-dimensional spaces in Chapter 8. In Chapter 9 we introduce an alternative approach to bounding the maximum degree of Steiner minimal trees from above, based on the illumination problem from combinatorial convexity. Finally, in Chapter 10 we consider the related k-Steiner minimal trees, which are shortest Steiner trees in which the number of Steiner points is restricted to be at most k.

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Jun 6, 2014

Konrad J. Swanepoel