  We consider the problem of estimating the mean of a random vector based on
$N$ independent, identically distributed observations. We prove the existence
of an estimator that has a near-optimal error in all directions in which the
variance of the one dimensional marginal of the random vector is not too small:
with probability $1-\delta$, the procedure returns $\wh{\mu}_N$ which satisfies
that for every direction $u \in S^{d-1}$, \[ \inr{\wh{\mu}_N - \mu, u}\le
\frac{C}{\sqrt{N}} \left( \sigma(u)\sqrt{\log(1/\delta)} + \left(\E\|X-\EXP
X\|_2^2\right)^{1/2} \right)~, \] where $\sigma^2(u) = \var(\inr{X,u})$ and $C$
is a constant. To achieve this, we require only slightly more than the
existence of the covariance matrix, in the form of a certain moment-equivalence
assumption.
  The proof relies on novel bounds for the ratio of empirical and true
probabilities that hold uniformly over certain classes of random variables.
