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


[1] DOI: 10.1080/02331939208843838 · Zbl 0815.26004
[2] DOI: 10.1137/0327055 · Zbl 0686.52006
[3] Phelps R.R., Convex Functions Monotone Operators and Dzyerentiability (1989)
[4] DOI: 10.1007/BF01195979 · Zbl 0798.49024
[5] DOI: 10.1007/BF02193061 · Zbl 0831.90105
[6] DOI: 10.1080/02331939508844125 · Zbl 0839.90092
[7] DOI: 10.1023/A:1022611314673 · Zbl 0886.90178
[8] DOI: 10.1007/BF02190127 · Zbl 0868.26005
[9] Rockafellar R.T., Convex Analysis · Zbl 0193.18401
[10] Söllner B., Eigenschaften {\(\gamma\)}-grobkonvexer Mengen und Funktionen (1991)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.