  Dependencies of the optimal constants in strong and weak type bounds will be
studied between maximal functions corresponding to the Hardy--Littlewood
averaging operators over convex symmetric bodies acting on $\mathbb R^d$ and
$\mathbb Z^d$. Firstly, we show, in the full range of $p\in[1,\infty]$, that
these optimal constants in $L^p(\mathbb R^d)$ are always not larger than their
discrete analogues in $\ell^p(\mathbb Z^d)$; and we also show that the equality
holds for the cubes in the case of $p=1$. This in particular implies that the
best constant in the weak type $(1,1)$ inequality for the discrete
Hardy--Littlewood maximal function associated with centered cubes in $\mathbb
Z^d$ grows to infinity as $d\to\infty$, and if $d=1$ it is equal to the largest
root of the quadratic equation $12C^2-22C+5=0$. Secondly, we prove
dimension-free estimates for the $\ell^p(\mathbb Z^d)$ norms, $p\in(1,\infty]$,
of the discrete Hardy--Littlewood maximal operators with the restricted range
of scales $t\geq C_q d$ corresponding to $q$-balls, $q\in[2,\infty)$. Finally,
we extend the latter result on $\ell^2(\mathbb Z^d)$ for the maximal operators
restricted to dyadic scales $2^n\ge C_q d^{1/q}$.
