Asymptotic behaviour of self-contracted planar curves and gradient   orbits of convex functions
  We hereby introduce and study the notion of self-contracted curves, which encompasses orbits of gradient systems of convex and quasiconvex functions. Our main result shows that bounded self-contracted planar curves have a finite length. We also give an example of a convex function defined in the plane whose gradient orbits spiral infinitely many times around the unique minimum of the function. 
Aris Daniilidis (LMPT), Olivier Ley (LMPT), St\'ephane Sabourau (LMPT)