## Symmetrically $$\gamma$$-convex functions.(English)Zbl 0943.26028

A real-valued function $$f$$ defined on a nonempty convex subset $$D$$ of a normed linear space is said to be symmetrically $$\gamma$$-convex ($$\gamma$$ being a strictly positive number) if for all $$x_0,x_1\in D$$ satisfying $$\|x_0- x_1\|> \gamma$$ one has $$f((1- \lambda)x_0+ \lambda x_1)\leq (1-\lambda)f(x_0)+ \lambda f(x_1)$$ for $$\lambda= \gamma/\|x_1- x_0\|$$. The authors prove that these functions enjoy nice analytical properties; in particular, when the space is finite-dimensional a symmetrically $$\gamma$$-convex function is locally Lipschitzian at each point which is at a distance greater than $$\gamma$$ from the boundary of the domain.

### MSC:

 26B25 Convexity of real functions of several variables, generalizations 26A51 Convexity of real functions in one variable, generalizations 26E15 Calculus of functions on infinite-dimensional spaces 49J45 Methods involving semicontinuity and convergence; relaxation

### Keywords:

$$\gamma$$-convex function; generalized convexity
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### References:

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