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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.

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
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