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 $f:\Delta \subset {\mathbb{R}}^k\to \mathbb{R}$ where $\Delta$ is a convex compact subset of ${\mathbb{R}}^k$. In the case when $\Delta$ 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 $\Delta$. 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.