Modified Delaunay empty sphere condition in the problem of approximation of the gradient

The classical Schwarz example shows that piecewise-linear approximation of smooth functions does not necessary yield convergence of the derivatives. However, in the planar case, the required convergence holds if the triangulation of the grid satisfies the empty sphere condition (that is, it is a Delaunay triangulation). These results do not extend to the multidimensional case, as is shown by our published examples. We give a modified empty sphere condition that also guarantees the necessary approximation in the multidimensional case.