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

## TL;DR

This paper investigates Min-Max affine approximants of convex or concave functions on convex sets, extending Chebyshev equioscillation results and connecting to computer graphics rendering techniques.

## Contribution

It proves the uniqueness of best affine approximants for convex functions on simplices and extends Chebyshev's theorem to higher dimensions.

## Key findings

- Unique best affine approximant exists for convex functions on simplices.
- Extension of Chebyshev equioscillation theorem to higher dimensions.
- Connections to efficient rendering of projective transformations in graphics.

## Abstract

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.