A New Notion of Path Length in the Plane

A path in the plane is a continuous function $\gamma$ from the unit interval into the plane. The Euclidean length of a path in the plane, when defined, has the following properties: it is invariant under isometries of the plane, it is monotone with respect to subpaths, and for any two points in the plane the path mapping homeomorphically onto the straight line segment joining them is the unique, nowhere constant path (up to homeomorphism) of minimal length connecting them. However, the Euclidean length does not behave well with respect to limits -- even when a sequence of paths $\gamma_i$ converges uniformly to a path $\gamma$ it is not necessarily true that their lengths converge. Moreover, for many paths the Euclidean length is not defined (i.e.\ not finite). In this paper we introduce an alternative notion of length of a path, $\length$, which has the above three properties, and is such that the length of any path is defined and finite. This enables one to compare the efficiency (shortness) of two paths between a given pair of points, even if their Euclidean lengths are infinite. In addition, if a sequence of paths $\gamma_i$ converges uniformly to a path $\gamma$, them $\lim \length(\gamma_i) = \length(\gamma)$. Moreover, this notion of length is defined for any map $\gamma$ from a locally connected continuum into the plane. We apply this notion of length to obtain a characterization of those families of paths which can be reparameterized to be equicontinuous. We also prove the existence of a unique shortest path in the closure of a given homotopy class of paths in an arbitrary plane domain, which extends known results for simply connected domains.