Orthogonal projection and minimal linear interpolation on a $n$-dimensional cube

Let $H$ be the orthogonal projection onto polynomials of $n$ variables of degree $\le 1$ and $\|\cdot\|$ be the norm of an operator from $C([0,1]^n)$ to $C([0,1]^n)$. In this paper we show that $C_1\theta_n\le\|H\|\le C_2\theta_n$, $n\in\mathrm{N}$. Here $\theta_n$ denotes the minimal norm of a projection dealing with the linear interpolation on the cube $[0,1]^n$. The proofs make use of certain properties of the Eulerian numbers and the central $B$-splines and also some previous results of the author.