The local stability of convexity, affinity and of the Jensen equation

Summary.

Let $C_D, A_D, J_D$ denote the smallest constants involved in the stability of convexity, affinity and of the Jensen equation of functions defined on a convex subset D of ${\Bbb R}^n$. By a theorem of J. W. Green, $C_D \le c\cdot \log (n+1)$ for every convex $D\subset {\Bbb R}^n$, where c is an absolute constant. We prove that the lower estimate $C_D \ge c\cdot \log (n+1)$ is also true, supposing that int $D \neq {\not 0}$.¶We show that $A_D \le 2 C_D$ and $A_D \le J_D \le 2A_D$ for every convex $D\subset {\Bbb R}^n$. The constant $J_D$ is not always of the same order of magnitude as $C_D$; for example $J_D = 1$ if $D ={\Bbb R}^n$. We prove that there are convex sets (e.g. the n-dimensional simplex) with $J_D \ge c\cdot \log n$.