Marstrand type projection theorems for normed spaces

TL;DR

This paper extends Marstrand projection theorems to certain normed spaces in , showing they hold for smooth norms but fail for norms with corners, and characterizes unrectifiable sets.

Contribution

It establishes Marstrand type theorems for normed spaces with smooth norms and provides a characterization of unrectifiable sets using transversality.

Findings

Projection theorems hold for smooth norms in 

Projection theorems fail for norms with corners

Characterization of unrectifiable sets via transversality

Abstract

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.