2013/01/25 by Hoehn, L. C., Oversteegen, L. G., Tymchatyn, E. D.
#54F50 #FOS: Mathematics #General Topology (math.GN) #Primary 54F15 #Secondary 54C20
paper · doi:10.48550/arxiv.1301.6070
For sufficiently tame paths in ℝn, Euclidean length provides a canonical parametrization of a path by length. In this paper we provide such a parametrization for all continuous paths. This parametrization is based on an alternative notion of path length, which we call len. Like Euclidean path length, len is invariant under isometries of ℝn, is monotone with respect to sub-paths, and for any two points in ℝn the straight line segment between them has minimal len length. Unlike Euclidean path length, the len length of any path is defined (i.e., finite) and len is continuous relative to the uniform distance between paths. We use this notion to obtain characterizations of those families of paths which can be reparameterized to be equicontinuous or compact. Finally, we use this parametrization to obtain a canonical homeomorphism between certain families of arcs.