*Preprint*

**Inserted:** 20 feb 2018

**Last Updated:** 1 dec 2018

**Journal:** Proc. AMS

**Year:** 2018

**Abstract:**

In this paper we prove that any $C^{1,\frac{1}{2}}$ curve in $\mathbb{R^n}$, is the solution of the gradient flow equation for some $C^1$ convex function $f$, if and only if it is strongly self-contracted.

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