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It should be straightforward to verify that $\mathcal A$ consists exactly of the lower semi-continuous radial and decreasing functions (which are non-negative and vanish at $\infty$): You already know that every $f\in\mathcal A$ has these properties, so you only have to verify that if $f$ has these properties, then $f=f^*$. If the definition is of any value, this should be rather straightforward to do (start with the classical case $n=1$).