  Consider the problem of estimating the mean of a Gaussian random vector when
the mean vector is assumed to be in a given convex set. The most natural
solution is to take the Euclidean projection of the data vector on to this
convex set; in other words, performing "least squares under a convex
constraint." Many problems in modern statistics and statistical signal
processing theory are special cases of this general situation. Examples include
the lasso and other high-dimensional regression techniques, function estimation
problems, matrix estimation and completion, shape-restricted regression,
constrained denoising, linear inverse problems, etc. This paper presents three
general results about this problem, namely, (a) an exact computation of the
main term in the estimation error by relating it to expected maxima of Gaussian
processes (existing results only give upper bounds), (b) a theorem showing that
the least squares estimator is always admissible up to a universal constant in
any problem of the above kind and (c) a counterexample showing that least
squares estimator may not always be minimax rate-optimal. The result from part
(a) is then used to compute the error of the least squares estimator in two
examples of contemporary interest.
