On the stability of Wright-convex functions

Summary.

We show that if D is an open and convex subset of $\mathbb{R}^N$ and $f: D \to \mathbb{R}$ fulfils the inequality¶¶ $f(tx + (1 - t)y) + f((1 - t)x+ty) \le f(x) + f(y) + 2\delta, \qquad x,y \in D, \quad t \in [0,1], \eqno{(\ast)}$¶¶ where $\delta \ge 0$ is a given constant, then there exists a Wright-convex function $g : D \to \mathbb{R}$ (i.e. g satisfies condition (*) for all $x,y \in D$ and $t \in [0,1]$ with $\delta = 0)$ such that¶¶ $\mid f(x) - g(x)\mid \le \theta\cdot\delta,\qquad x \in D,$¶¶ where $\theta$ is a constant depending only on the dimension N.