Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 18 (2011), No. 2, 391--396Copyright Heldermann Verlag 2011 Semiconcave Functions with Power Moduli Jacek Tabor Institute of Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348 Kraków, Poland tabor@ii.uj.edu.pl Józef Tabor Institute of Mathematics, University of Rzeszów, Rejtana 16A, 35-959 Rzeszów, Poland tabor@univ.rzeszow.pl Anna Murenko Institute of Mathematics, University of Rzeszów, Rejtana 16A, 35-959 Rzeszów, Poland aniam@univ.rzeszow.pl [Abstract-pdf] A function $f$ is approximately convex if $$f(\alpha x+(1-\alpha )y)\leq \alpha f(x)+(1-\alpha)f(y) + R(\alpha, \| x-y\|),$$ 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, Hamilton-Jacobi Equations, and Optimal Control", Birkh\"{a}user, Boston 2004]. We improve an estimation involved in such relation in the above-mentionded book and show that our result is sharp. Keywords: Semiconcave function, paraconvex function, Jensen convexity, modulus of semiconcavity. MSC: 26B25; 39B82 [ Fulltext-pdf  (97  KB)] for subscribers only.