We introduce the isoperimetry coefficient σ(G) = |∂G|n/(n−1)/|G| of region G ⊂ Rn. In terms of this the error δΔ(f) estimates for the gradient of the piecewise linear interpolation of functions of class C1(G), C2(G), C1,α(G), 0 < α < 1, are obtained. The problem of obtaining such estimates is nontrivial, especially in the multidimensional case. Here it should be noted that in the two-dimensional case, for functions of class C2(G), the convergence of the derivatives is provided by the classical Delaunay condition.
