\L ojasiewicz-type inequalities and global error bounds for nonsmooth definable functions in o-minimal structures

TL;DR

This paper establishes new lasiewicz-type inequalities and conditions for global error bounds for continuous definable functions within o-minimal structures, linking these bounds to the Palais-Smale condition.

Contribution

It introduces novel lasiewicz-type inequalities and characterizes when global error bounds exist for nonsmooth definable functions, connecting these bounds to the Palais-Smale condition.

Findings

Derived lasiewicz-type inequalities for definable functions

Provided necessary and sufficient conditions for global error bounds

Linked global error bounds to the Palais-Smale condition

Abstract

In this paper, we give some {\L}ojasiewicz-type inequalities and a nonsmooth slope inequality on non-compact domains for continuous definable functions in an o-minimal structure. We also give a necessary and sufficicent condition for which global error bound exists. Moreover, we point out the relationship between the Palais-Smale condition and this global error bound.