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

S. J. Dilworth, Ralph Howard, James W. Roberts

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…