On a multidimensional analogue of the Schwarz example

We use the Delaunay triangulation to construct a series of examples showing that in the multidimensional case, in contrast to the two-dimensional case, the approximation property for derivatives fails to hold for piecewise-linear approximations of a smooth function. These examples are closely related to the classical Schwarz example concerning piecewise-linear approximations of the lateral surface of a right circular cylinder.