On trajectories of steepest descent for non-smooth functions

In 1980 DeGiorgi, Marino and Tosques used the newly introduced concept of "slope" of a function on a metric space (which is the maximal instantaneous speed of decrease of the function at a given point) to define trajectories of "maximal slope" as curves whose metric derivative coincides with the slope almost everywhere. Subsequently DeGiorgi himself, Marino, Ambrosio and others proved, under certain additional assumptions the existence of curves with a slightly weaker property that can be called trajectories of "near maximal slope". (A detailed up to date account of available results can be found in the monograph by L. Ambrosio, N. Gigli, and G. Savare. Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zurich. Birkhauser Verlag, Basel, second edition, 2008.)
We propose a new strategy for constructing such trajectories. It is based on principles of metric regularity theory (in which, as has been recently understood, slopes play a fundamental role) and allows to get all mentioned results with less efforts and in a way that seems to be geometrically very transparent. The approach is especially efficient for functions on Euclidean spaces. In this case, if the function is locally Lipschitz and has good structural properties (e.g. semi-algebraic or, more generally, definable in a certain o-minimal structure – such functions are typical in practical optimization), its trajectories of near-maximal slope satisfy a certain evolution inclusion associated with the limiting subdifferential. Moreover, it turns out that the lengths of all such trajectories lying within a certain bounded region are bounded by the same constant (depending on the region).