On definable multifunctions and {\L}ojasiewicz inequalities

We investigate several possibilities of obtaining a {\L}ojasiewicz 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 {\L}ojasiewicz-type inequalities for definable multifunctions obtained from Clarke’s subgradient or the tangent cone. The paper ends with a {\L}ojasiewicz-type subgradient inequality in the spirit of Bolte-Daniilidis-Lewis-Shiota or Ph\d{a}m.