On Min-Max Affine Approximants of Convex or Concave Real-Valued Functions from ℝ^k, Chebyshev Equioscillation and Graphics

We study min-max affine approximants of a continuous convex or concave function f:Δ⊆ℝk→ℝ, where Δ is a convex compact subset of ℝ^k. In the case when Δ is a simplex, we prove that there is a vertical translate of the supporting hyperplane in ℝ^k+1 of the graph of f at the vertices which is the unique best affine approximant to f on Δ. For k = 1, this result provides an extension of the Chebyshev equioscillation theorem for linear approximants. Our result has interesting connections to the computer graphics problem of rapid rendering of projective transformations.