Using convex integration we give a constructive proof of the well-known fact that every continuous curve in a contact 3-manifold can be approximated by a Legendrian curve

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N. Hungerbühler, T. Mettler, M. Wasem. “Convex Integration and Legendrian Approximation of Curves.” Journal of Convex Analysis 24 (2017), No. 1, 309–317.