Let $a,b\in\mathbb{R}$, $a. Consider \begin{align} C[a,b] & :=\{f\in\mathbb{R}^{[a,b]}:f\text{ is continuous}\}\text{,} \\ L_{\infty}[a,b] & :=\{f\in\mathbb{R}^{[a,b]}:f\text{ is Lebesgue-Borel measurable and essentially bounded}\} \end{align} equipped with the norm \begin{equation} \DeclareMathOperator{\esssup}{ess sup} C[a,b]\ni f \mapsto\|f\|_{C[a,b]}:=\sup_{x\in[a,b]}|f(x)| \end{equation} and the seminorm \begin{equation} L_{\infty}[a,b]\ni f \mapsto\|f\|_{L_{\infty}[a,b]}:=\esssup_{x\in[a,b]}|f(x)|\text{,} \end{equation} respectively. Moreover, for $n\in\mathbb{N}_0:=\{0,1,2,\dots\}$, $\Pi_n[a,b]$ denotes the linear space of real algebraic polynomial functions of degree at most $n$ restricted to $[a,b]$.
The following statement is well known as Chebyshev Equioscillation Theorem:
$P\in\Pi_n[a,b]$ is a best approximation from $\Pi_n[a,b]$ to $f\in C[a,b]$, that is, \begin{equation} \|P-f\|_{C[a,b]} =\inf_{Q\in\Pi_n[a,b]}\|Q-f\|_{C[a,b]} \end{equation} if and only if there exists an increasing sequence of points $x_i\in[a,b]$, $0\le i\le n+1$, and $\sigma\in\{-1,1\}$ such that \begin{equation} P(x_i)-f(x_i) =\sigma\ (-1)^i\|P-f\|_{C[a,b]}\text{,$\quad0\le i\le n+1$.}\tag{*} \end{equation}
Does a similar statement hold if $C[a,b]$ is replaced by $L_{\infty}[a,b]$?
I am particularly interested in the case $n=1$ and the implication \begin{equation} \text{$P$ is a best approximation to $f$}\implies\text{A relation similar to (*) holds.} \end{equation}
Clearly, (*) cannot remain unchanged in the generalized setting for if $P\in\Pi_n[a,b]$ is a best approximation from $\Pi_n[a,b]$ to $f\in L_{\infty}[a,b]$ then $P$ is a best approximation from $\Pi_n[a,b]$ to each $g\in L_{\infty}[a,b]$ such that $\|g-f\|_{L_{\infty}[a,b]}=0$.