
Journal of Convex Analysis 18 (2011), No. 2, 391396 Copyright Heldermann Verlag 2011 Semiconcave Functions with Power Moduli Jacek Tabor Institute of Computer Science, Jagiellonian University, Lojasiewicza 6, 30348 Kraków, Poland tabor@ii.uj.edu.pl Józef Tabor Institute of Mathematics, University of Rzeszów, Rejtana 16A, 35959 Rzeszów, Poland tabor@univ.rzeszow.pl Anna Murenko Institute of Mathematics, University of Rzeszów, Rejtana 16A, 35959 Rzeszów, Poland aniam@univ.rzeszow.pl [Abstractpdf] A function $f$ is approximately convex if $$ f(\alpha x+(1\alpha )y)\leq \alpha f(x)+(1\alpha)f(y) + R(\alpha, \ xy\), $$ for $x,y \in \mathrm{dom} f$, $\alpha\in [0,1]$ and for a respective perturbation term $R$. If the above inequality is assumed only for $\alpha=\frac{1}{2}$, then the function $f$ is called Jensen approximately convex.\par The relation between Jensen approximate convexity and approximate convexity has been investigated in many papers, in particular for semiconcave functions [see P. Cannarsa and C. Sinestrari, "Semiconcave Functions, HamiltonJacobi Equations, and Optimal Control", Birkh\"{a}user, Boston 2004]. We improve an estimation involved in such relation in the abovementionded book and show that our result is sharp. Keywords: Semiconcave function, paraconvex function, Jensen convexity, modulus of semiconcavity. MSC: 26B25; 39B82 [ Fulltextpdf (97 KB)] for subscribers only. 