We consider the problem of recovering a high-dimensional vector observed in white noise, where the vector is known to be sparse, but the degree of sparsity is unknown. We consider three different ways of defining sparsity of a vector: using the fraction of nonzero terms; imposing power-law decay bounds on the ordered entries; and controlling the norm for p small. We obtain a procedure which is asymptotically minimax for loss, simultaneously throughout a range of such sparsity classes. The simultaneous asymptotic minimaxity is achieved by a data-adaptive thresholding scheme, based on controlling the False Discovery Rate (FDR). FDR control is a recent innovation in simultaneous testing, in which one seeks to ensure that at most a certain fraction of the rejected null hypotheses will correspond to false rejections. In our treatment, the FDR control parameter q plays an informative role in understanding how to achieve asymptotic minimaxity. Our results say that letting with problem size n is sufficient for asymptotic minimaxity, while keeping q > 1/2 prevents asymptotic minimaxity. To our knowledge, this relation between ideas in simultaneous inference and asymptotic decision theory is new. Our work suggest insights about a class of model selection procedures which has been introduced recently by several authors. These new procedures are based on complexity penalization of the form . We exhibit a close connection to FDR-controlling procedures with q tending to 0, which strongly supports a conjecture of simultaneous asymptotic minimaxity of such procedures.