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

Abstract

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.

Kurdyka-Łojasiewicz functions and mapping cylinder neighborhoods

“…The article by Bolte, Daniilidis, Ley, and Mazet [2], a benchmark on the topic of KŁ functions, and the references therein, will give the reader a broader view of their alternative characterizations and applications. The recently found counterexample to the Thom Gradient Conjecture [5] for the class of KŁ functions only increases the intrigue surrounding them.…”