Geodesic
Semiconcave Functions with Power Moduli
Journal of convex analysis, Tome 18 (2011) no. 2, pp. 391-396.

Voir la notice de l'article provenant de la source Heldermann Verlag

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.
Classification : 26B25, 39B82
Mots-clés : Semiconcave function, paraconvex function, Jensen convexity, modulus of semiconcavity 
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     author = {J. Tabor and J. Tabor and A. Murenko},
     title = {Semiconcave {Functions} with {Power} {Moduli}},
     journal = {Journal of convex analysis},
     pages = {391--396},
     publisher = {mathdoc},
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     number = {2},
     year = {2011},
     url = {http://geodesic.mathdoc.fr/item/JCA_2011_18_2_JCA_2011_18_2_a5/}
}
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J. Tabor; J. Tabor; A. Murenko. Semiconcave Functions with Power Moduli. Journal of convex analysis, Tome 18 (2011) no. 2, pp. 391-396. http://geodesic.mathdoc.fr/item/JCA_2011_18_2_JCA_2011_18_2_a5/