On Long Orbit Empty Value (LOEV) principle

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 ${\mathrm{\Sigma }}_g$-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.