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

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Let 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 f(∑ᵢ₌₁ᵇ tᵢxᵢ) ≤ ∑ᵢ₌₁ᵇ tᵢf(xᵢ) +1 for all x₁,...,xB ∈ Δₙ and all (t₁,...,tB)∈ ΔB₋₁. We determine explicitly the extremal (i.e. pointwise largest) 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.