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

AbstractLet n⩾1 and B⩾2. A real-valued function f defined on the n-simplex Δn is approximately convex with respect to ΔB−1 iff∑i=1Btixi⩽∑i=1Btif(xi)+1 for all x1,…,xB∈Δn and all (t1,…,tB)∈ΔB−1. We determine the extremal function of this type which vanishes on the vertices of Δn. 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