Let $P_n$ be the vector space of real multivariate polynomials in $d$ variables of total degree no more than $n$ and let $f \in C(X)$, where $X \subset \mathbb{R}^d$ is compact. Let $(p_n)_{n=1}^\infty$ be a sequence of the best $L_1$ approximations of $f$ from $P_n$, i.e.,
$$ p_n \in \underset{p \in P_n}{\mathrm{argmin}}\; \| p - f \|_{L_1(X)} . $$
Does this sequence converge uniformly to $f$ on $X$?. If not in general, what further assumptions have to be made on $f$ or $X$ in order for uniform convergence to take place?
