Menger curvature and Lipschitz parametrizations in metric spaces
Volume 185 / 2005
Fundamenta Mathematicae 185 (2005), 143-169
MSC: Primary 51F99; Secondary 30E20.
DOI: 10.4064/fm185-2-3
Abstract
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 $\Omega(\varepsilon)$, the class of bounded metric spaces $E$ such that the maximum angle for every triple in $E$ is at least $\pi/2 + \arcsin\varepsilon$. Finally, we extend Peter Jones's travelling salesman theorem to general metric spaces.
