Behavior of the gradient flow: the convex case

The classical Lojasiewicz inequality and its extension to o-minimal
structures by K. Kurdyka has a considerable impact on the analysis of
gradient-like methods and related problems. In this talk we shall discuss
alternative characterizations of this type of inequality via the notion of
a defragmented gradient curve: such curves have uniformly bounded lengths
if and only if the Kurdyka-Lojasiewicz inequality is satisfied. Another
characterization in terms of talweg lines will be given. In the convex case
these results are significantly reinforced, allowing in particular to
establish a kind of asymptotic equivalence for discrete gradient methods
and continuous gradient curves.