  We prove that in a metric measure space $X$, if for some $p \in (1,\infty)$
there are uniform bounds (independent of the measure) for the weak type $(p,p)$
of the centered maximal operator, then $X$ satisfies a certain geometric
condition, the Besicovitch intersection property, which in turn implies the
uniform weak type $(1,1)$ of the centered operator. Thus, the following
characterization is obtained: the centered maximal operator satisfies uniform
weak type $(1,1)$ bounds if and only if the space $X$ has the Besicovitch
intersection property.
  In $\mathbb{R}^d$ with any norm, the constants coming from the Besicovitch
intersection property are bounded above by the translative kissing numbers. The
extensive literature on kissing numbers allows us to obtain, first, sharp
estimates on the uniform bounds satisfied by the centered maximal operators
defined by arbitrary norms on the plane, second, sharp estimates in every
dimension when the $\ell_\infty$ norm is used, and third, improved estimates in
all dimensions when considering euclidean balls, as well as the sharp constant
in dimension 3.
  Additionally, we prove that the existence of uniform $L^1$ bounds for the
averaging operators associated to arbitrary measures and radii, is equivalent
to a weaker variant of the Besicovitch intersection property.
