# Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities

@article{Semmes1996FindingCO,
title={Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincar{\'e} inequalities},
author={S. Semmes},
journal={Selecta Mathematica},
year={1996},
volume={2},
pages={155-295}
}
• S. Semmes
• Published 1 December 1996
• Mathematics
• Selecta Mathematica
In many metric spaces one can connect an arbitrary pair of points with a curve of finite length, but in Euclidean spaces one can connect a pair of points with a lot of rectifiable curves, curves that are well distributed across a region. In the present paper we give geometric criteria on a metric space under which we can find similar families of curves. We shall find these curves by first solving a “dual” problem of building Lipschitz maps from our metric space into a sphere with good…
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