Geometric dilation of closed curves in normed planes
type: Item
instance of: scholarly article
title: Geometric dilation of closed curves in normed planes
zbMATH Open document ID: 1161.65014
author: Senlin Wu
author: Horst Martini
published in: Computational Geometry
publication date: 2009-03-09T00:00:00Z
review text: The detour between two points \(p\) and \(q\) on a rectifiable simple closed curve, \(C\), in the Euclidean plane, is the ratio between the length of a shortest path connecting \(p\) and q in \(C\) and their Euclidean distance \(|pq|\). The maximum detour over all pairs of points is called the geometric dilation. The authors extend this definition to (normed or) Minkowski planes and obtain lower bounds for the geometric dilation of rectifiable simple curves on them. It is known that the lower bound for the geometric dilation of rectifiable simple closed curves in the Euclidean plane is \(\pi/2\), which can be attained only by circles [cf. \textit{A. Ebbers-Baumann, A. Grüne} and \textit{R. Klein}, Comput. Geom. 37, No.~3, 188--208 (2007; Zbl 1121.65020) or \textit{A. Dumitrescu, A. Ebbers-Baumann, A. Grüne, R. Klein} and \textit{G. Rote}, Comput. Geom. 36, No.~1, 16--38 (2007; Zbl 1115.65018)]. This result is extended to Minkowski planes \(X\): the lower bound is analogously a quarter of the circumference of the unit circle \(SX\) of \(X\), but can also be attained by curves that are not Minkowskian circles. The proof of the main result is based on the study of the relation between the intrinsic distances of a curve and of the boundary of its convex hull. In addition it is shown that the lower bound is attained only by Minkowskian circles if the respective norm is strictly convex.
reviewed by: Juan Monterde
Mathematics Subject Classification ID: 65D18
zbMATH DE Number: 5526413
zbMATH Keywords: arc length
zbMATH Keywords: convex curve
zbMATH Keywords: convex geometry
zbMATH Keywords: detour
zbMATH Keywords: geometric dilation
zbMATH Keywords: halving pair
zbMATH Keywords: Minkowski plane
zbMATH Keywords: normed plane
zbMATH Keywords: rectification
zbMATH Keywords: strictly convex norm
zbMATH Keywords: Minkowskian circles
MaRDI profile type: Publication
full work available at URL: j.comgeo.2008.06.001
OpenAlex ID: W1968950445
cites work: Computing the detour and spanning ratio of paths, trees, and cycles in 2D and 3D
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cites work: Q4112910
cites work: Q4716272
cites work: On Closed Curves in Minkowski Spaces
DOI: 10.1016/J.COMGEO.2008.06.001
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description: scientific article; zbMATH DE number 5526413