Compactly Epi-lipschitzian Convex Sets and Functions in Normed Spaces

The concept of compactly epi-Lipschitzian (CEL) sets in locally convex topological spaces was introduced by Borwein and Strojwas [6]. It is an extension of Rockafellar’s concept of epi-Lipschitzian sets [36]. An advantage of the CEL property is that it always holds in finite dimensional spaces and, in contrast to its epi-Lipschitzian predecessor, makes it possible to recapture much of the detailed information available in finite dimensions. The original motivation for introducing the CEL concept was to select class of closed sets in infinite dimensions (primarily in Banach spaces) for which the Clarke tangent and normal cones [11] adequately measure boundary behavior. A number of strong results were obtained in this direction; see [3], [6], [7], [8], and references therein. At the same time it was clarified that the CEL property is not sufficient for the (weak-star) locally compactness of the Clarke normal cone at boundary points [3, Example 4.1]. To get ∗Research supported by NSERC and by the Shrum endowment of Simon Fraser University for the first author, research partly supported by the Pacific Institute for the Mathematical Sciences for the second author, and research partly supported by the National Science Foundation under grant DMS-9704751 and by an NSERC Foreign Researcher Award for the third author.