A counterexample concerning the $L_2$-projector onto linear spline spaces

For the $L_2$-orthogonal projection $P_V$ onto spaces of linear splines over simplicial partitions in polyhedral domains in $\mathbb{R}^d$, $d>1$, we show that in contrast to the one-dimensional case, where $\Vert P_V\Vert _{L_\infty\to L_\infty} \le 3$ independently of the nature of the partition, in higher dimensions the $L_\infty$-norm of $P_V$ cannot be bounded uniformly with respect to the partition. This fact is folklore among specialists in finite element methods and approximation theory but seemingly has never been formally proved.