Shortest paths in arbitrary plane domains
Keyword(s):
Open Set
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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.
1965 ◽
Vol 17
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pp. 373-382
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Keyword(s):
Keyword(s):
2010 ◽
Vol 348
(9-10)
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pp. 521-524
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2005 ◽
Vol 139
(1)
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pp. 149-159
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