We establish Marstrand-type as well as Besicovich-Federer-type projection theorems for closest-point projections onto hyperplanes in the normed space $\mathbb{ R}^{ n}$. In particular, we prove that if a norm on $\mathbb{ R}^{ n}$ is $C^{ 1,1}$-regular, then the analogues of the well-known statements from the Euclidean setting hold. On the other hand, we construct an example of a $C^{ 1}$-regular norm in $\mathbb{ R}^2$ for which Marstrand-type theorems fail.
Categories: math.MG