Semiconcave Functions with Power Moduli

A function

is approximately convex if

f(\alpha x+(1-\alpha )y)\leq \alpha f(x)+(1-\alpha)f(y) + R(\alpha, \| x-y\|), f ( α x + ( 1 − α ) y ) ≤ α f ( x ) + ( 1 − α ) f ( y ) + R ( α , ∥ x − y ∥ ) ,

for

x,y \in \mathrm{dom} f x , y ∈ d o m f

,

\alpha\in [0,1] α ∈ [ 0 , 1 ]

and for a respective perturbation term

R R

. If the above inequality is assumed only for

\alpha=\frac{1}{2} α = 1 2

, then the function

f f

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.