Geometric dilation of closed curves in normed planes
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.
scientific article; zbMATH DE number 5526413
scientific article