  The classical Lojasiewicz inequality and its extensions for partial
differential equation problems (Simon) and to o-minimal structures (Kurdyka)
have a considerable impact on the analysis of gradient-like methods and related
problems: minimization methods, complexity theory, asymptotic analysis of
dissipative partial differential equations, tame geometry. This paper provides
alternative characterizations of this type of inequalities for nonsmooth lower
semicontinuous functions defined on a metric or a real Hilbert space. In a
metric context, we show that a generalized form of the Lojasiewicz inequality
(hereby called the Kurdyka-Lojasiewicz inequality) relates to metric regularity
and to the Lipschitz continuity of the sublevel mapping, yielding applications
to discrete methods (strong convergence of the proximal algorithm). In a
Hilbert setting we further establish that asymptotic properties of the semiflow
generated by $-\partial f$ are strongly linked to this inequality. This is done
by introducing the notion of a piecewise subgradient curve: such curves have
uniformly bounded lengths if and only if the Kurdyka-Lojasiewicz inequality is
satisfied. Further characterizations in terms of talweg lines -a concept linked
to the location of the less steepest points at the level sets of $f$- and
integrability conditions are given. In the convex case these results are
significantly reinforced, allowing in particular to establish the asymptotic
equivalence of discrete gradient methods and continuous gradient curves. On the
other hand, a counterexample of a convex C^2 function in in the plane is
constructed to illustrate the fact that, contrary to our intuition, and unless
a specific growth condition is satisfied, convex functions may fail to fulfill
the Kurdyka-Lojasiewicz inequality.
