Marstrand-type projection theorems for linear projections and in normed spaces

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.