## Faculty Works

#### Title

On a nonconvex separation theorem and the approximate extremal principle in Asplund spaces.

#### Document Type

Conference Paper in Published Proceedings

#### Department

Mathematics and Computer Science

2013

#### Abstract

In this paper we establish a revised version of the {\em nonconvex separation theorem} established in [J.M. Borwein and A. Jofr\'{e}, A nonconvex separation property in Banach spaces, Math. Meth. Oper. Res. {\bf 48} 169--179 (1996)]. Our new separation theorem is formulated in terms of Fr\'echet normal cones in Asplund spaces while theirs was formed with an abstract kind of generalized differentiation enjoying a good calculus. It is, indeed, shown to be equivalent to the known {\em approximate extremal principle} in [B. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Grundlehren Series (Fundamental Principles of Mathematical Sciences) {\bf 330}, Springer, Berlin (2006)]. In addition, we discuss several efficient conditions ensuring the extremality property of a system of nonconvex sets.