  In this paper we address the following question: given $ p\in (1,\infty)$, $n
\geq 1$, does there exists a constant $A(p,n)>1$ such that $\| M
f\|_{L^{p}}\geq A(n,p) \| f\|_{L^{p}}$ for any nonnegative $f \in
L^{p}(\mathbb{R}^{n})$, where $Mf$ is a maximal function operator defined over
the family of shifts and dilates of a centrally symmetric convex body. The
inequality fails in general for the centered maximal function operator, but
nevertheless we give an affirmative answer to the question for the uncentered
maximal function operator and the almost centered maximal function operator. In
addition, we also present the Bellman function approach of Melas, Nikolidakis
and Stavropoulos to maximal function operators defined over various types of
families of sets, and in case of parallelepipeds we will show that
$A(n,p)=\left(\frac{p}{p-1}\right)^{1/p}$.
