On $\Phi $-Convexity in Extremal Problems

For a class of functions $\Phi $ on an arbitrary set X, $\Phi $-convex subsets of X and functions on X are defined, the latter being least upper bounds of some functions from $\Phi $. Also the generalized Fenchel transform and $\Phi $-subgradients are determined and their properties investigated. $\Phi $-convexity and $\Phi $-subdifferentiability of lower-semicontinuous functions on metric spaces are examined with respect to special important families $\Phi $. Among related results, we present a theorem on the existence of minimizing points of nonlinear functions on Banach spaces and extensions of the notion of Hölder continuity. The relevance of the theory to perturbed extremal problems is indicated.