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Shortest paths in arbitrary plane domains

Part of: Real and complex geometry Special properties

Published online by Cambridge University Press:  09 November 2020

L. C. Hoehn ,
L. G. Oversteegen  and
E. D. Tymchatyn
L. C. Hoehn
Affiliation:
Department of Computer Science & Mathematics, Nipissing University, 100 College Drive, Box 5002, North Bay, ON P1B 8L7, Canada e-mail: loganh@nipissingu.ca
L. G. Oversteegen*
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA
E. D. Tymchatyn
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK S7N 5E6, Canada e-mail: tymchat@math.usask.ca
*
e-mail: overstee@uab.edu
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Abstract

Let $\Omega $ be a connected open set in the plane and $\gamma : [0,1] \to \overline {\Omega }$ a path such that $\gamma ((0,1)) \subset \Omega $ . We show that the path $\gamma $ can be “pulled tight” to a unique shortest path which is homotopic to $\gamma $ , via a homotopy h with endpoints fixed whose intermediate paths $h_t$ , for $t \in [0,1)$ , satisfy $h_t((0,1)) \subset \Omega $ . We prove this result even in the case when there is no path of finite Euclidean length homotopic to $\gamma $ under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a “shortest” path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.

Keywords

Pathlengthshortestplane domainhomotopyanalytic covering map

MSC classification

Primary: 54F50: Spaces of dimension $leq 1$; curves, dendrites
Secondary: 51M25: Length, area and volume
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 2 , April 2022 , pp. 349 - 367
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

L.C.H. was partially supported by NSERC grant RGPIN 435518. L.G.O. was partially supported by NSF-DMS-1807558. E.D.T. was partially supported by NSERC grant OGP-0005616.

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