= Solution
Let $u^*\in\mathcal U$, $e=f-u^*$, $d=\|e\|_\infty$, and $E=\{x:|e(x)|=d\}$. The real <Kolmogorov criterion for uniform approximation> says \b[$u^*$ is best if and only if]
$$
\boxed{\text{for every }v\in\mathcal U,\quad\min_{x\in E}e(x)v(x)\leq0.}
$$
Equivalently, no direction $v$ has the error's sign strictly at every extremal point. If $d=0$, the criterion is automatic.
For sufficiency, a point with $e(x)v(x)\leq0$ gives $|e(x)-v(x)|\geq d$, so $u^*+v$ cannot improve the norm. For necessity, suppose $e(x)v(x)>0$ on $E$. Compactness gives a positive lower bound there and on a neighbourhood of $E$. On that neighbourhood, $(e-\tau v)^2=e^2-2\tau ev+\tau^2v^2<d^2$ for sufficiently small positive $\tau$. Off the neighbourhood, $|e|$ has a strict gap below $d$, and small $\tau$ preserves that gap. Thus $u^*+\tau v$ has smaller error, a contradiction. This also explains why the criterion involves all active extrema.
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