  We consider Marstrand type projection theorems for closest-point projections
in the normed space $\mathbb{R}^2$. We prove that if a norm on $\mathbb{R}^2$
is regular enough, then the analogues of the well-known statements from the
Euclidean setting hold, while they fail for norms whose unit balls have
corners. We establish our results by verifying Peres and Schlag's
transversality property and thereby also obtain a Besicovitch-Federer type
characterization of purely unrectifiable sets.
