  We prove that the Galerkin finite element solution $u_h$ of the Laplace
equation in a convex polyhedron $\varOmega$, with a quasi-uniform tetrahedral
partition of the domain and with finite elements of polynomial degree $r\ge 1$,
satisfies the following weak maximum principle: \begin{align*}
\left\|u_{h}\right\|_{L^{\infty}(\varOmega)} \le
C\left\|u_{h}\right\|_{L^{\infty}(\partial \varOmega)} , \end{align*} with a
constant $C$ independent of the mesh size $h$. By using this result, we show
that the Ritz projection operator $R_h$ is stable in $L^\infty$ norm uniformly
in $h$ for $r\geq 2$, i.e. \begin{align*} \|R_hu\|_{L^{\infty}(\varOmega)} \le
C\|u\|_{L^{\infty}(\varOmega)} . \end{align*} Thus we remove a logarithmic
factor appearing in the previous results for convex polyhedral domains.
