On Long Orbit Empty Value (LOEV) principle

TL;DR

This paper explores the LOEV principle in Variational Analysis, proving new results and characterizing semicompleteness for generalized metric functions, with applications to perturbability in metric spaces.

Contribution

It introduces novel applications of the LOEV principle, including characterizations of semicompleteness for non-symmetric, non-triangle inequality metric functions.

Findings

Characterization of $\\Sigma_g$-semicompleteness via Ekeland Theorem

New results in Variational Analysis using LOEV principle

Application to perturbability in $G_\delta$ subsets of complete metric spaces

Abstract

We consider an useful in Variational Analysis tool -- Long Orbit or Empty Value (LOEV) principle -- in different settings, starting from more abstract to more defined. We prove, using LOEV principle, a number of basic results in Variational Analysis, including some novel. We characterize $\Sigma_g$-semicompleteness for a generalized metric function $g$ which is neither symmetric nor satisfies the triangle inequality, in terms of validity of Ekeland Theorem for this $g$. We present an interesting application to perturbability to minimum in a $G_\delta$ subset of a complete metric space.