Menger curvature and Lipschitz parametrizations in metric spaces

Volume 185 / 2005

Immo Hahlomaa Fundamenta Mathematicae 185 (2005), 143-169 MSC: Primary 51F99; Secondary 30E20. DOI: 10.4064/fm185-2-3


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.


  • Immo HahlomaaDepartment of Mathematics and Statistics
    University of Jyväskylä
    P.O. Box 35
    FIN-40014 Jyväskylä, Finland

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