Almost approximately convex functions.(English)Zbl 0641.39006

After defining “negligible sets”, a common generalization of sets of Haar measure 0 and of Baire sets of first category (meager sets) for groups [see J. G. Dhombres and the author Glas. Mat., III. Ser. 13(33), 39-62 (1978; Zbl 0384.39004)], the author proves, among others, roughly the following result. Let D be a convex subset of $${\mathbb{R}}^ n$$. If, for an $$\epsilon >0$$ and for all $$\delta\in [0,1]$$, f:D$$\to {\mathbb{R}}$$ satisfies $$f(\delta x+(1-\delta)y)\leq \delta f(x)+(1- \delta)f(y)+\epsilon$$ on $$D^ 2$$, with the possible exception of a negligible set, then there exists a convex g:D$$\to {\mathbb{R}}$$ such that $$| f(x)-g(x)| <c\epsilon$$ (c depends only upon n) on D, again with the possible exception of a negligible set. Somewhat weaker results are presented on more general linear spaces or groups and/or for midpoint-convex functions.
There is an unusually large number of misprints for any acceptable standard and, in particular, for this author. Some are of little consequence, others droll (e.g., p. 63, l. 25: “equal to something equals something.”) and several seriously impede understanding (e.g., p. 64, l. 18; p. 70, l. 5).
Reviewer: J.Aczél

MSC:

 39B72 Systems of functional equations and inequalities 16L30 Noncommutative local and semilocal rings, perfect rings 28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures 46A03 General theory of locally convex spaces 03E15 Descriptive set theory

Zbl 0384.39004
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