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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.

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
Full Text: DOI
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