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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 γ-convexity and the inner γ-convexity 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 γ-convex functions and inner γ-convex ones.
For instance, each γ-minimizer of an outer γ-convex 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 γ-convex 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 γ-extreme point of D,
which cannot be represented as midpoint of some segment $[z', z'']\subset
D$ with $\|z'-z''\|\geq 2\,\gamma$, etc.
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Keywords: Generalized convexity, rough convexity,
outer γ-convex function, inner γ-convex function, perturbation of
convex function, self-Jung constant, γ-extreme point.
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