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

Abstract

Let n⩾1 and B⩾2. A real-valued function f defined on the n-simplex Δ n is approximately convex with respect to Δ B−1 if f ∑ i=1 B t ix i ⩽ ∑ i=1 B t if(x i)+1 for all x 1,…, x B ∈ Δ n and all ( t 1,…, t B )∈ Δ B−1 . We determine the extremal function of this type which vanishes on the vertices of Δ 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 ε-convex functions.