Semiconcave Functions with Power Moduli

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.