Marstrand type projection theorems for normed spaces

We consider Marstrand type projection theorems for closest-point projections in the normed space ℝ². We prove that if a norm on ℝ² 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 set