The Steiner ratio for the dual normed plane
type: Item
instance of: scholarly article
title: The Steiner ratio for the dual normed plane
zbMATH Open document ID: 0888.05015
DOI: 10.1016/S0012-365X(96)00080-5
author: Peng-Jun Wan
author: Ron Graham
author: Du Ding-Zhu
published in: Discrete Mathematics
publication date: 1998-06-01T00:00:00Z
review text: The Steiner ratio \(\rho\) of the real plane \(\mathbb{R}^2\) is the infimum of \(L_S(X)\cdot L_M(X)^{-1}\) taken over all finite subsets \(X\) of \(\mathbb{R}^2\), where \(L_S(X)\) resp. \(L_M(X)\) denotes the total length of a minimum Steiner tree resp. a minimum spanning tree of \(X\). If the length is measured in view of a metric given by a compact, convex, centrally symmetric domain \(D\) instead of the circular unit disc then one gets a Steiner ratio \(\rho(D)\) depending on \(D\). One of the authors (together with others) has conjectured in a previous paper [see Discrete Comput. Geom. 9, No. 4, 351-370 (1993; Zbl 0774.05028)] that \(\rho(D)= \rho(D^*)\) where \(D^*\) denotes the polar body of \(D\). In this paper it is shown that \(\rho_5(D)= \rho_5(D^*)\) with \(\rho_n\) denoting the Steiner ratio taken only of subsets \(X\) consisting of at most five points.
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 05C35
zbMATH DE Number: 1047094
zbMATH Keywords: Steiner ratio
zbMATH Keywords: Steiner tree
zbMATH Keywords: spanning tree
zbMATH Keywords: polar body
reviewed by: Gerd Wegner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1047094