The local stability of convexity, affinity and of the Jensen equation
Let \(D \subset \mathbb R^n\) be a convex set. A function \(f:D \to \mathbb R\) is said to be \(\delta\)-convex if \[ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\delta, \quad x,y \in D, \quad t \in [0,1]. \] A function \(f:D \to \mathbb R\) is said to be \(\delta\)-affine if \[ |f(tx+(1-t)y)-tf(x)-(1-t)f(y)|\leq \delta, \quad x,y \in D, \quad t \in [0,1] \] and it is \(\delta\)-Jensen if \[ \Big|f\Big(\frac{x+y}{2}\Big)-\frac{f(x)+f(y)}{2}\Big|\leq \delta, \quad x,y \in D. \] The following stability theorem holds. {Theorem} i) If \(f\) is \(\delta\)-convex, then there exists a convex function \(g:D \to \mathbb R\) such that \(|f(x)-g(x)|\leq C\delta\), \(x \in D\); ii) If \(f\) is \(\delta\)-affine, then there exists an affine function \(k:\mathbb R^n\to \mathbb R\) such that \(|f(x)-k(x)|\leq A\delta\), \(x \in D\); iii) If \(f\) is \(\delta\)-Jensen, then there exists a Jensen function \(h:\mathbb R^n \to \mathbb R\) such that \(|f(x)-h(x)|\leq J\delta\), \(x \in D\). The constants \(C, A, J\) depend only on \(D\). If we denote by \(C_D, A_D, J_D\) the infima of the constants in the above theorem, the author proves that: \[ C_D \geq c\log(n+1), \quad \text{if } \int D \neq \emptyset,\qquad A_D \leq 2C_D,\qquad A_D \leq J_D \leq 2A_D. \]
scientific article; zbMATH DE number 1346981