# zbMATH — the first resource for mathematics

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
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.