α-Covex Sets and Strong Quasiconvexity

A major property of continuous strong convex functions is that they are inf-compact; hence, their minima are reached. In the past, several attempts have been made to extend strong convexity to generalized convexity, but they fail in the sense that they do not ensure inf-compactness.

In this paper, we introduce a concept of strong convexity on sets from which we deduce new concepts of strong quasiconvexity related to inf-compactness.