Abstract

The classical Lojasiewicz inequality asserts that for every critical point $a$ of a $C1$ subanalytic function $f$ there exists such that the quantity $|f-f(a)|\theta |\nabla \; f|-1$ remains bounded around $a$. This relatively simple fact is extremely important, since it ensures the finiteness of the lengths of all bounded trajectories of the gradient system $x\prime (t)=-\nabla \; f(x(t))$ (thus guaranteeing in particular their convergence). Recently, K. Kurdyka extended the aforementioned result for $C1$ functions belonging to arbitrary o-minimal structures. Namely, for every definable $C1$ function $f:U\to \; R$₊ ($U$ being an open bounded subset of $Rn$ )\; there\; exist$ \rho ,c>0$and\; a\; strictly\; increasing\; definable\; function$ \psi :(0,\rho )\to (0,+\infty )$ of\; class$ C1$ such\; that$ ||\nabla (\psi \; o\; f)(x)||\ge \; c,$for\; each$ x\in \; f-1(0,\rho )$.\; This\; Kurdyka-Lojasiewicz\; inequality\; is\; thus\; satisfied\; by\; a\; much\; larger\; class\; of\; functions\; and\; turns\; out\; to\; be\; sufficient\; to\; ensure\; the\; finiteness\; of\; the\; bounded\; gradient\; trajectories.\; We\; shall\; endeavor\; to\; shed\; light\; on\; this\; relation\; by\; characterizing\; intrinsically\; the\; fact\; that\; the\; lengths\; of\; the\; bounded\; trajectories.of\; the\; gradient\; system\; are\; uniformly\; bounded.\; The\; case\; of\; a\; convex\; function$ f$is\; particularly\; challenging\; and\; remains\; open.$$$$$$