Interpolation by polyhedral functions

A polyhedral function $l_{P(\Delta_n)}(f)$. interpolating a function $f$, defined on a polygon $\Phi$, is defined by a set of interpolating nodes $\Delta_n\subset\Phi$ and a partition $P(\Delta_n)$ of the polygon $\Phi$ into triangles with vertices at the points of $\Delta_n$. In this article we will compute for convex moduli of continuity the quatities
$$E(H_\Phi^\omega;P(\Delta_n))=\sup_{f\in H_\Phi^\omega}\|f-l_{P(\Delta_n)}(f)\|,$$
and also give an asymptotic estimate of the quantities
$$E_n(H_\Phi^\omega)=\inf_{\Delta_n}\inf_{P(\Delta_n)}E(H_\Phi^\omega;P(\Delta_n)).$$