  This paper is devoted to proving the general {\L}ojasiewicz inequality, in
both the definable and subanalytic cases, under the most relaxed assumptions.
It means that we drop the usual continuity and compactness assumptions. In the
second part of the paper we concentrate on the {\L}ojasiewicz inequality for
multifunctions and apply it to the natural multifunctions related to the medial
axis of a set (basic notion in pattern recognition).
