On the stability of almost convex functions

Abstract:

Let ${\mathbf {R}}$ denote the set of real numbers and $I$ an open interval of ${\mathbf {R}}$. A function $f:I \to R$ is said to be almost $\delta$-convex iff $f(tx + (1 - t)y) \leq tf(x) + (1 - t)f(y) + \delta$ holds for all $(x,y) \in I \times I\backslash N$, where $N \subset I \times I$ is of measure zero, each $t \in [0,1]$ and some $\delta \geq 0$. It is proved that such a function is uniformly close to a convex function almost everywhere.