an:06864725
Zbl 1387.26029
Vesel??, Libor; Zaj????ek, Lud??k
A non-DC function which is DC along all convex curves
EN
J. Math. Anal. Appl. 463, No. 1, 167-175 (2018).
00387011
2018
j
26B25 26A51
DC function; d.c. function; characterization of DC functions
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 \(\mathbb{R}^n\), \(n > 1\).