## Extremal approximately convex functions and the best constants in a theorem of Hyers and Ulam

**A 2000 Preprint by
S. Dilworth,
R. Howard, and
J. Roberts
**

- 2000:26
Let $n\geq1$ and $B\geq2$. A real-valued function

*f*defined on the*n-*simplex $\Delta _ n$ is approximately convex with respect to $\Delta _ {B-1}$ if$$f\left(\sum^B _ {i=1}t _ ix _ i\right)\leq\sum^B _ {i=1}t _ if(x _ i)+1$$

for all $x _ 1,\ldots,x _ B\in\Delta _ n$ and all $(t _ 1,\ldots,t _ B)\in\Delta _ {B-1}$. We determine the extremal 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.