  We prove that in a metric measure space $(X, d, \mu)$, the averaging
operators $A_{r, \mu }$ satisfy a uniform strong type $(1,1)$ bound $\sup_{r,
\mu} \|A_{r, \mu }\|_{L^1\to L^1} < \infty$ if and only if $X$ satisfies a
certain geometric condition, the equal radius Besicovitch intersection
property.
