  We study central limit theorems for the projected sample mean of independent
and identically distributed observations on subsets $\mathcal Q \subset \mathbb
R^2$ of the Euclidean plane.
  It is well-known that two conditions suffice to obtain a parametric rate of
convergence for the projected sample mean: $\mathcal Q$ is a $\mathcal
C^2$-manifold, and the expectation of the underlying distribution calculated in
$\mathbb R^2$ is bounded away from the medial axis, the set of point that do
not have a unique projection to $\mathcal Q$.
  We show that breaking one of these conditions can lead to any other rate: For
a virtually arbitrary prescribed rate, we construct $\mathcal Q$ such that all
distributions with expectation at a preassigned point attain this rate.
