Modified Delaunay empty sphere condition in the problem of approximation of the gradient (Q2821885)

Let \(P = \{p_j \in {\mathbb R}^n:\,j=1,\dots,n\}\) be given and let \(T\) be a triangulation of \(P\). If \(S\in T\) is a simplex with the vertices \(p_{j_k}\), \(k=0,\dots,n\), then \(f_S\) is the linear function with \(f(p_{j_k}) = f_S(p_{j_k})\), \(k=0,\dots,n\). Then the value \(\nabla f_S(x)\) is used as approximation of the gradient \(\nabla f(x)\) for all \(x\in S\). A triangulation \(T\) is called a Delaunay triangulation if the empty sphere condition holds, i.e., for every simplex \(S\in T\), no point of \(P\) lies inside its circumscribed sphere. \textit{V. A. Klyachin} and \textit{A. A. Shirokii} [Russ. Math. 56, No. 1, 27--34 (2012); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2012, No. 1, 31--39 (2012; Zbl 1254.68338)] have shown for \(n=2\) and for triangulations of \(\varepsilon\)-networks \(P\) that the error fulfills the estimate NEWLINE\[NEWLINE \max_{S\in T} \left( \max_{x\in S} | \nabla f (x) - \nabla f_S (x)|\right) \leq C(f)\, \varepsilon NEWLINE\]NEWLINE with a constant \(C(f)\) depending on \(f\). But for \(n\geq 3\), the empty sphere condition is not sufficient for such an estimate.NEWLINENEWLINEIn this paper, the author gives a modified empty sphere condition that guarantees such an estimate for arbitrary dimension \(n\geq 3\).