  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 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.
