A convex function satisfying the Lojasiewicz inequality but failing the gradient conjecture both at zero and infinity

Aris Daniilidis (1) , Mounir Haddou (2) , Olivier Ley (2)

1 CMM - Center for Mathematical Modeling
2 IRMAR - Institut de Recherche Mathématique de Rennes

We construct an example of a smooth convex function on the plane with a strict minimum at zero, which is real analytic except at zero, for which Thom's gradient conjecture fails both at zero and infinity. More precisely, the gradient orbits of the function spiral around zero and at infinity. Besides, the function satisfies the Lojasiewicz gradient inequality at zero.

Mots clés: Gradient conjecture; gradient conjecture at infinity; Kurdyka-Lojasiewicz inequality; convex function; convergence of secants

Domaines: Optimisation et contrôle [math.OC]; Systèmes dynamiques [math.DS]

Aris Daniilidis, Mounir Haddou, Olivier Ley. A convex function satisfying the Lojasiewicz inequality but failing the gradient conjecture both at zero and infinity. Bulletin of the London Mathematical Society, 2022, ⟨10.1112/blms.12586⟩. ⟨hal-03136550v2⟩