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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.

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