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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 \(\mathbb{R}^n\), \(n > 1\).
26B25 Convexity of real functions of several variables, generalizations
26A51 Convexity of real functions in one variable, generalizations
Full Text: DOI
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