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Characterization of convex functions. (English) Zbl 1167.26003

There are a lot of linear inequalities in the literature that are known to be valid for all convex functions, but also to characterize convexity. The paper provides a tool for showing “automatically” results of the kind: if a given nontrivial linear inequality is true for all continuous convex functions, then every continuous function satisfying it, is convex. The tool in question is based on the following result, which is interesting by itself: Let \(K\subseteq \mathbb{R}^{n}\) be a compact convex set, \(g:K\rightarrow \mathbb{R}\) a fixed nonconvex continuous function and \(\varepsilon>0\). Then every continuous function \(h:K\rightarrow \mathbb{R}\) can be uniformly approximated by functions of the form \(f+\sum_{i=1}^{k} \lambda_{i}g\circ a_{i}\), where \(f:K\rightarrow \mathbb{R}\) is continuous convex, \(a_{i}:K\rightarrow K\) (\(i=1,\ldots k\), \(k\in \mathbb{N}\)) are affine with Lipschitz constant less than \(\varepsilon\), and \(\lambda_{i}\geq0\).

MSC:

26B25 Convexity of real functions of several variables, generalizations
26D15 Inequalities for sums, series and integrals
39B62 Functional inequalities, including subadditivity, convexity, etc.
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