Approximation error for linear polynomial interpolation on \(n\)-simplices (Q1387360)

Let \(\Delta_n \subset \mathbb{R}^n\) be an \(n\)-simplex and let \(W^2_nM\) be the class of functions \(f:\Delta_n \to\mathbb{R}\) having continuous second derivatives not exceeding \(M> 0\) in absolute value. The author gives an (exact) pointwise bound for the error \(| f(x)-P_{1,n} (f,x) |\), where \(P_{1,n} (f,x)\) denotes the linear polynomial interpolating \(f\in W^2_nM\) at the vertices of the simplex \(\Delta_n\). More precisely, he shows that \[ \sup \biggl\{ \bigl| f(x)-P_{1,n} (f,x)\bigr|: f\in W^2_n M\biggr\} =\textstyle {1\over 2} M\bigl( R^2_n-d^2_n (x)\bigr), \] where \(R_n\) is the radius of the sphere circumscribing \(\Delta_n\) and \(d_n(x)\) denotes the distance between \(x\) and the center of this sphere. As a corollary he obtains a uniform bound in terms of the Chebyshev radius of the simplex \(\Delta_n\).