Geodesic
Shortest paths in arbitrary plane domains
Canadian journal of mathematics, Tome 74 (2022) no. 2, pp. 349-367

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DOI

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.
DOI : 10.4153/S0008414X20000784
Mots-clés : Path, length, shortest, plane domain, homotopy, analytic covering map 
Hoehn, L. C.; Oversteegen, L. G.; Tymchatyn, E. D. Shortest paths in arbitrary plane domains. Canadian journal of mathematics, Tome 74 (2022) no. 2, pp. 349-367. doi: 10.4153/S0008414X20000784
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     title = {Shortest paths in arbitrary plane domains},
     journal = {Canadian journal of mathematics},
     pages = {349--367},
     year = {2022},
     volume = {74},
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     doi = {10.4153/S0008414X20000784},
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