Journal of Convex Analysis 18 (2011), No. 2, 391--396
Copyright 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
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