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 $f:\Delta\subset \mathbb R^k\xrightarrow{} \mathbb R$ 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 verticies 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.