Abstract

A function ff is approximately convex if f(αx+(1−α)y)≤αf(x)+(1−α)f(y)+R(α,∥x−y∥),f(\alpha x+(1-\alpha )y)\leq \alpha f(x)+(1-\alpha)f(y) + R(\alpha, \| x-y\|), for x,y∈domfx,y \in \mathrm{dom} f, α∈[0,1]\alpha\in [0,1] and for a respective perturbation term RR. If the above inequality is assumed only for α=12\alpha=\frac{1}{2}, then the function ff is called Jensen approximately convex. 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", Birkhäuser, Boston 2004]. We improve an estimation involved in such relation in the above-mentionded book and show that our result is sharp.

Author information

Contact details are reproduced from the original publication and may be historical.

Jacek Tabor

Institute of Computer Science, Jagiellonian University, Łojasiewicza 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 Mureńko

Institute of Mathematics, University of Rzeszów, Rejtana 16A, 35-959 Rzeszów, Poland

aniam@univ.rzeszow.pl

Keywords

  • Semiconcave function
  • paraconvex function
  • Jensen convexity
  • modulus of semiconcavity

Mathematics Subject Classification

26B25; 39B82

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Suggested citation

J. Tabor, J. Tabor, A. Mureńko. “Semiconcave Functions with Power Moduli.” Journal of Convex Analysis 18 (2011), No. 2, 391–396. https://doi.org/10.68381/jca18022

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.