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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 \(\mathbb{R}^n\), \(n > 1\).
Reviewer: Reviewer (Berlin)
MSC:
26B25 Convexity of real functions of several variables, generalizations
26A51 Convexity of real functions in one variable, generalizations
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References:
[1] Bačák, M.; Borwein, J., On difference convexity of locally Lipschitz functions, Optimization, 60, 961-978, (2011) · Zbl 1237.46007
[2] Duda, J., Curves with finite turn, Czechoslovak Math. J., 58, 23-49, (2008) · Zbl 1167.46321
[3] Duda, J.; Veselý, L.; Zajíček, L., On d.c. functions and mappings, Atti Semin. Mat. Fis. Univ. Modena, 51, 111-138, (2003) · Zbl 1072.46025
[4] Prudnikov, I. M., On the question of the representability of a function of two variables as the difference of convex functions, Sibirsk. Mat. Zh., Sib. Math. J., 55, 6, 1116-1125, (2014), translation in · Zbl 1320.26012
[5] Tuy, H., Convex analysis and global optimization, Springer Optimization and Its Applications, vol. 110, (2016), Springer · Zbl 1362.90001
[6] Veselý, L.; Zajíček, L., Delta-convex mappings between Banach spaces and applications, Dissertationes Math., 289, (1989), 52 pp · Zbl 0685.46027
[7] Veselý, L.; Zajíček, L., On vector functions of bounded convexity, Math. Bohem., 133, 321-335, (2008) · Zbl 1199.47242
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