A characterization of the uniform strong type $(1,1)$ bounds for averaging operators

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 $\sup_{r, \mu} |A_{r, \mu }|_{L^1\to L^1} < \infty$ifandonlyif$X$ satisfies a certain geometric condition, the equal radius Besicovitch intersection property.