We show that pointwise bounds on the Menger curvature imply Lipschitz parametrization for general compact metric spaces. We also give some estimates on the optimal Lipschitz constants of the parametrizing maps for the metric spaces in Ω(ε), the class of bounded metric spaces E such that the maximum angle for every triple in E is at least π/2 + arcsinε. Finally, we extend Peter Jones's travelling salesman theorem to general metric spaces.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm185-2-3,
author = {Immo Hahlomaa},
title = {Menger curvature and Lipschitz parametrizations in metric spaces},
journal = {Fundamenta Mathematicae},
volume = {185},
year = {2005},
pages = {143-169},
zbl = {1077.54016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm185-2-3}
}
Immo Hahlomaa. Menger curvature and Lipschitz parametrizations in metric spaces. Fundamenta Mathematicae, Tome 185 (2005) pp. 143-169. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm185-2-3/