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A non-DC function which is DC along all convex curves. (English) Zbl 1387.26029
Summary: A problem asked by the authors in 1989 concerns the natural question, whether one can deduce that a continuous function $$f$$ on an open convex set $$D \subset \mathbb{R}^n$$ is DC (i.e., is a difference of two convex functions) from the behavior of $$f$$ “along some special curves $$\varphi$$ ”. I. M. Prudnikov published in 2014 a theorem (working with convex curves $$\varphi$$ in the plane), which would give a positive answer in $$\mathbb{R}^2$$ to our problem. However, in the present note we construct an example showing that this theorem is not correct, and thus our problem remains open in each $$\mathbb{R}^n$$, $$n > 1$$.
##### MSC:
 26B25 Convexity of real functions of several variables, generalizations 26A51 Convexity of real functions in one variable, generalizations
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##### References:
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