On Min-Max affine approximants of convex or concave real valued functions from $mathbb R^k$, Chebyshev equioscillation and graphics

We study Min-Max affine approximants of a continuous convex or concave function $f:Deltasubset mathbb R^kxrightarrow{} mathbb R$ 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.