On Min-Max affine approximants of convex or concave real valued functions from $\mathbb R^k$, Chebyshev equioscillation and graphics
This work addresses a theoretical problem in approximation theory with incremental contributions, potentially benefiting computer graphics by improving rendering efficiency.
The paper tackles the problem of finding the best affine approximants for convex or concave functions on simplices in ℝᵏ, proving that a vertical translate of the supporting hyperplane at the vertices is the unique best approximant. For k=1, this extends the Chebyshev equioscillation theorem, with applications to rapid rendering in computer graphics.
We study Min-Max affine approximants of a continuous convex or concave function $f:Δ\subset \mathbb R^k\xrightarrow{} \mathbb R$ 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.