  Adaptive estimation of linear functionals over a collection of parameter
spaces is considered. A between-class modulus of continuity, a geometric
quantity, is shown to be instrumental in characterizing the degree of
adaptability over two parameter spaces in the same way that the usual modulus
of continuity captures the minimax difficulty of estimation over a single
parameter space. A general construction of optimally adaptive estimators based
on an ordered modulus of continuity is given. The results are complemented by
several illustrative examples.
