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

Abstract

We study min-max affine approximants of a continuous convex or concave function $f:\Delta \subseteq \mathbb R^k\xrightarrow {} \mathbb R$, where Δ is a convex compact subset of $\mathbb R^k$. In the case when Δ is a simplex, we prove that there is a vertical translate of the supporting hyperplane in $\mathbb R^{k+1}$ 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.