# Self-contracted curves are gradient flows of convex functions

## TL;DR

This paper establishes a characterization of certain smooth curves in Euclidean space as gradient flows of convex functions, linking geometric properties with variational principles.

## Contribution

It proves that strongly self-contracted $C^{1,eta}$ curves are exactly the gradient flows of convex functions, providing a new geometric-analytic characterization.

## Key findings

- Strongly self-contracted $C^{1,eta}$ curves are gradient flows of convex functions.
- Characterization bridges geometric properties with convex analysis.
- Results apply to curves with $eta ext{ in } (rac{1}{2},1]$.

## Abstract

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