## 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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### References:

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