Let $n\ge 1$ be an integer. $\mathcal Q_n$ be the set of all polynomial functions over $[a,b]$, of degree exactly $n$.
My question is : Is it true that
$\inf_{x_0,x_1,...,x_n\in[a,b], x_0<x_1<...<x_n} \sup_{x\in [a,b]} \prod_{i=0}^n |(x-x_i)|=\inf_{Q\in \mathcal Q_n } \sup_{x\in [a,b]} |x^{n+1}-Q(x)|$ ?
I can easily see that $\inf_{x_0,x_1,...,x_n\in[a,b],x_0<x_1<...<x_n} \sup_{x\in [a,b]} \prod_{i=0}^n |(x-x_i)| \ge \inf_{Q\in \mathcal Q_n } \sup_{x\in [a,b]} |x^{n+1}-Q(x)|$ , but I don't know about the other reverse inequality.
Please help.