  This paper provides a discrete Poincar\'e inequality in $n$ space dimensions
on a simplex $K$ with explicit constants. This inequality bounds the norm of
the piecewise derivative of functions with integral mean zero on $K$ and all
integrals of jumps zero along all interior sides by its Lebesgue norm by
$C(n)\operatorname{diam}(K)$. The explicit constant $C(n)$ depends only on the
dimension $n=2,3$ in case of an adaptive triangulation with the newest vertex
bisection. The second part of this paper proves the stability of an enrichment
operator, which leads to the stability and approximation of a (discrete)
quasi-interpolator applied in the proofs of the discrete Friedrichs inequality
and discrete reliability estimate with explicit bounds on the constants in
terms of the minimal angle $\omega_0$ in the triangulation. The analysis allows
the bound of two constants $\Lambda_1$ and $\Lambda_3$ in the axioms of
adaptivity for the practical choice of the bulk parameter with guaranteed
optimal convergence rates.
