The following theorem can be found in many lectures on approximation theory:
Theorem: Let $f$ be a continuous function defined on the interval $[a,b]$. Then there exists a polynomial $p_n\in\Pi_n$ that minimizes $\max_{a\le x\le b}\|f(x)-p_n(x)\|$ among all $p_n\in\Pi_n$, where $\Pi_n$ is the set of polynomials of degree $n$.
My question now is: Does an analogous theorem exist in the multivariate case, i.e. for $f$ defined on a compact subset of $\mathbb{R}^d$? And if this is the case: Can the resulting multivariate polynomial explicitly be calculated (similar to the Remez algorithm in dimension 1)?