Approximation error for linear polynomial interpolation onn-simplices

LetW _n ² M be the class of functionsf: Δ _n → ℝ (when Δ _n is ann-simplex) with bounded second derivative (whose absolute value does not exceedM>0) along any direction at an arbitrary point of the simplex Δ _n . LetP _1,n (f;x) be the linear polynomial interpolatingf at the vertices of the simplex. We prove that there exists a functiong ∈ W _n ² M such that for anyf ∈W _n ² M and anyx ∈ Δ _n one has |f (x)−P ₁, _n (f;x)|≤g(x). Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 504–510, October, 1996. I thank Yu. N. Subbotin for posing the problem and for his attention to my work.