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Generalized convexities of lower semicontinuous functions. (English) Zbl 0585.26008

Let \({\mathcal F}\) be a class of real-valued functions defined on a real interval I, and let \(L\subset I\) be also an interval. By using \({\mathcal F}\) the author defines three types of generalized convex functions as follows. A function \(g:L\to {\mathbb{R}}\) is said to be (1) \({\mathcal F}\)-convex on L if for all \(t',t''\in L,\) \(t'\neq t'',\) and each \(\lambda \in (0,1)\) there exists a function \(f\in {\mathcal F}\) satisfying the following conditions: \((C_ 1)\) f is continuous on \((t',t'');\) \((C_ 2)\quad f(t')=g(t'),f(t'')=g(t'');\) \((C_ 3)\quad g((1-\lambda)t'+\lambda t'')\leq f((1-\lambda)t'+\lambda t'');\) (2) \({\mathcal F}\)-convexlike if for all \(t',t''\in L,\) \(t'\neq t'',\) and at least one \(\lambda \in (0,1)\) there exists a function \(f\in {\mathcal F}\) satisfying the conditions \((C_ 1)\), \((C_ 2)\) and \((C_ 3)\); (3) uniformly \({\mathcal F}\)-convexlike if there is a \(\delta \in (0,1/2)\) such that for all \(t',t''\in L,\) \(t'\neq t'',\) and at least one \(\lambda \in [\delta,1-\delta]\) there exists a function \(f\in {\mathcal F}\) satisfying the conditions \((C_ 1)\), \((C_ 2)\) and \((C_ 3)\). The main result of the paper reveals conditions on \({\mathcal F}\) which assure that every \({\mathcal F}\)-convexlike (resp. uniformly \({\mathcal F}\)-convexlike) function \(g:L\to {\mathbb{R}}\) which is lower semicontinuous on the interior of L is \({\mathcal F}\)-convex on L.
Reviewer: W.W.Breckner

MSC:

26A51 Convexity of real functions in one variable, generalizations
26B25 Convexity of real functions of several variables, generalizations
90C25 Convex programming
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