Abstract. The most kinds of generalized
convexities cannot resist perturbations, even linear ones, while real
application problems are often affected by disturbances, both linear and
nonlinear ones. For instance, we showed earlier that quasiconvexity,
explicit quasiconvexity, and pseudoconvexity cannot withstand arbitrarily
small linear disturbances to keep their characteristic properties, and
convex functions are the only ones which can resist every linear disturbance
to preserve property “each local minimizer is a global minimize”, but it
fails if perturbation is nonlinear, even with arbitrarily small supremum
norm. In this paper, we present some sufficient conditions for the outer
γconvexity and the inner γconvexity of disturbed functions, for
instance, when convex functions are added with arbitrarily wild but
accordingly bounded functions. That means, in spite of such nonlinear
disturbances, some weakened properties can be saved, namely the properties
of outer γconvex functions and inner γconvex ones. For
instance, each γminimizer of an outer γconvex function f: D > R
defined by $f(x^*)=\inf_{x\in\bar B(x^*,\gamma)\cap D}f(x)$ is a global
minimizer, or if an inner γconvex function f: D > R defined
on some bounded convex subset D
of an inner product space attains its supremum, then it does so at least at
some strictly γextreme point of D,
which cannot be represented as midpoint of some segment $[z', z'']\subset
D$ with $\z'z''\\geq 2\,\gamma$, etc.

Keywords: Generalized convexity, rough convexity,
outer γconvex function, inner γconvex function, perturbation of
convex function, selfJung constant, γextreme point.
