On best uniform approximation of finite sets by linear combinations of real valued functions using linear programming

We study the best approximation problem: $${\displaystyle \underset{\alpha \in {\mathbb{R}}^m}{\min }\underset{1\le i\le n}{\max }\mid y_i-\sum _{j=1}^m{\alpha }_j{\mathrm{\Gamma }}_j({\mathbf{x}}_i)\mid \mathrm{.}}$$ Here: $\mathrm{\Gamma }:=\{{\mathrm{\Gamma }}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\mathrm{\Gamma }}_m\}$ is a list of functions where for each $1\le j\le m$, $\Gamma_j:\Delta \rightarrow \mathbb R$with$\Delta$asetofevaluationpoints$\left{{\bf x_1},…,{\bf x_n}\right}$\mathrm{.}$\left{y_1,…,y_n\right}$isasetofrealvaluesand$\mathbb R^m:=\left{(\alpha_1,…,\alpha_m),, \alpha_j\in \mathbb R,, 1\leq j\leq m\right}$.