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

TL;DR

This paper extends Marstrand-type projection theorems to normed spaces, showing they hold under certain smoothness conditions but can fail with less regular norms, thus broadening the understanding of geometric measure theory in different settings.

Contribution

It establishes Marstrand-type and Besicovich-Federer-type projection theorems for projections in normed spaces, identifying conditions for their validity and providing counterexamples.

Findings

Marstrand-type theorems hold for $C^{1,1}$-regular norms.

Counterexample shows failure of Marstrand-type theorems for some $C^{1}$-regular norms.

Results extend geometric measure theory to non-Euclidean normed spaces.

Abstract

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.