Approximation of the gradient of a function on the basis of a special class of triangulations

We introduce the class of $\Phi$-triangulations of a finite set $P$ of points in $\mathbb{R}^n$ analogous to the classical Delaunay triangulation. Such triangulations can be constructed using the condition of empty intersection of $P$ with the interior of every convex set in a given family of bounded convex sets the boundary of which contains the vertices of a simplex of the triangulation. In this case the classical Delaunay triangulation corresponds to the family of all balls in $\mathbb{R}^n$. We show how $\Phi$-triangulations can be used to obtain error bounds for an approximation of the derivatives of $C^2$-smooth functions by piecewise linear functions.