Isoperimetry coefficient for simplex in the problem of approximation of~derivatives

V. A. Klyachin ; D. V. Shurkaeva

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 2, pp. 151-160

Résumé

We introduce the isoperimetry coefficient $\sigma(G)= {|\partial G|^{n/(n-1)}}/{|G|} $ of region $G\subset \mathbb R^n$. In terms of this the error $\delta_\Delta(f)$ estimates for the gradient of the piecewise linear interpolation of functions of class $C^1(G)$, $C^2(G)$, $C^{1,\alpha}(G)$, $0\alpha1$, 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 $C^2(G)$, the convergence of the derivatives is provided by the classical Delaunay condition. In the multidimensional case, as shown by the examples, such conditions are not sufficient. Nevertheless, the article shows how to apply these estimates to the Delaunay triangulation of multidimensional discrete $ \varepsilon $-nets. The results obtained give sufficient conditions for convergence of the derivatives on the Delaunay triangulation of discrete $ \varepsilon $-nets with $ \varepsilon \to 0 $. In addition, the ratio of the distortion factor is found for isoperimetry coefficient under the quasi-isometric transformation.