On definable multifunctions and Łojasiewicz inequalities

Abstract

We investigate several possibilities of obtaining a Łojasiewicz inequality for definable multifunctions and give some examples of applications thereof. In particular, we prove that the Hausdorff distance and its extension to closed sets is definable when composed with definable multifunctions. This allows us to obtain Łojasiewicz-type inequalities for definable multifunctions obtained from Clarke's subgradient or the tangent cone. The paper ends with a Łojasiewicz-type subgradient inequality in the spirit of Bolte-Daniilidis-Lewis-Shiota or Ph\d{a}m.

Together with Paulina Pełszyńska we added a new section to the previous version of the preprint; namely, the part concerning the subgradient inequality. To appear in J. Math. Anal. Appl. (2017)