Extremal Approximately Convex Functions and the Best Constants in a Theorem of Hyers and Ulam

Abstract

Let $n\ge 1$ and $B\ge 2$. A real-valued function $f$ defined on the $n$-simplex ${\Delta }_n$ is approximately convex with respect to ${\Delta }_{B-1}$ iff f(\sum_{i=1}^B t_ix_i) \le \sum_{i=1}^B t_if(x_i) +1 for all $x_1,...,x_B\in {\Delta }_n$ and all $(t_1,...,t_B)\in {\Delta }_{B-1}$. We determine explicitly the extremal (i.e. pointwise largest) function of this type which vanishes on the vertices of ${\Delta }_n$. We also prove a stability theorem of Hyers-Ulam type which yields as a special case the best constants in the Hyers-Ulam stability theorem for $\epsilon$-convex functions.

12 pages 1 figure