  Recently, a vector version of Witsenhausen's counterexample was considered
and it was shown that in that limit of infinite vector length, certain
quantization-based control strategies are provably within a constant factor of
the optimal cost for all possible problem parameters. In this paper, finite
vector lengths are considered with the dimension being viewed as an additional
problem parameter. By applying a large-deviation "sphere-packing" philosophy, a
lower bound to the optimal cost for the finite dimensional case is derived that
uses appropriate shadows of the infinite-length bound. Using the new lower
bound, we show that good lattice-based control strategies achieve within a
constant factor of the optimal cost uniformly over all possible problem
parameters, including the vector length. For Witsenhausen's original problem --
the scalar case -- the gap between regular lattice-based strategies and the
lower bound is numerically never more than a factor of 8.
