Geodesic
Approximation error for linear polynomial interpolation on $n$-simplices
Matematičeskie zametki, Tome 60 (1996) no. 4, pp. 504-510

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Let $W_n^2M$ be the class of functions $f\colon\Delta_n\to\mathbb R$ (when ($\Delta_n$ is an $n$-simplex) with bounded second derivative (whose absolute value does not exceed $M>0$) along any direction at an arbitrary point of the simplex $\Delta_n$. Let $P_{1,n}(f;x)$ be the linear polynomial interpolating $f$ at the vertices of the simplex. We prove that there exists a function $g\in W_n^2M$ such that for any $f\in W_n^2M$ and any $x\in\Delta_n$ one has $$ |f(x)-P_{1,n}(f;x)|\leqslant g(x). $$ 
@article{MZM_1996_60_4_a2,
     author = {Yu. A. Kilizhekov},
     title = {Approximation error for linear polynomial interpolation on $n$-simplices},
     journal = {Matemati\v{c}eskie zametki},
     pages = {504--510},
     publisher = {mathdoc},
     volume = {60},
     number = {4},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_60_4_a2/}
}
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Yu. A. Kilizhekov. Approximation error for linear polynomial interpolation on $n$-simplices. Matematičeskie zametki, Tome 60 (1996) no. 4, pp. 504-510. http://geodesic.mathdoc.fr/item/MZM_1996_60_4_a2/