Approximation error for linear polynomial interpolation on $n$-simplices

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
$$ |f(x)-P_{1,n}(f;x)|\leqslant g(x). $$