# 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} }

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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