  We propose an estimator for the mean of random variables in separable real
Banach spaces using the empirical characteristic function. Assuming that the
covariance operator of the random variable is bounded in a precise sense, we
show that the proposed estimator achieves the optimal sub-Gaussian rate, except
for a faster decaying mean-dependent term. Under a mild condition, an iterative
refinement procedure can essentially eliminate the mean-dependent term and
provide a shift-equivariant estimate. Our results particularly suggests that a
certain Gaussian width that appears in the best known rate in the literature
might not be necessary. Furthermore, using the boundedness of the
characteristic functions, we also show that, except possibly at high
signal-to-noise ratios, the proposed estimator is gracefully robust against
adversarial "contamination". Our analysis is overall concise and transparent,
thanks to the tractability of the characteristic functions.
