  In the classical many normal means with different variances, we consider the
situation when the observer is allowed to allocate the available measurement
budget over the coordinates of the parameter of interest. The benchmark is the
minimax linear risk over a set. We solve the problem of optimal allocation of
observations under the measurement budget constrain for two types of sets,
ellipsoids and hyperrectangles. By elaborating on the two examples of Sobolev
ellipsoids and hyperectangles, we demonstrate how re-allocating the
measurements in the (sub-)optimal way improves on the standard uniform
allocation. In particular, we improve the famous Pinsker (1980) bound.
