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

Let $n\ge1$ and $B\ge2$. 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.