θ-metric function in the problem of minimization of functionals

We study approximative properties of sets as a function of the rate of variation of the distance function defined in terms of some continuous functional (in lieu of a metric). As an application, we prove non-uniqueness of approximation by non-convex subsets of Hilbert spaces with respect to special continuous functionals. Results of this kind are capable of proving non-uniqueness solvability for gradient-type equations.