# Direction sets, Lipschitz graphs and density

## TL;DR

This paper classifies subsets of Euclidean space based on their direction sets, revealing a trichotomy linked to whether the set is a Lipschitz graph, a non-Lipschitz graph, or not a graph, with implications for geometric measure theory.

## Contribution

It establishes a novel trichotomy categorizing subsets by their direction sets and explores consequences under various conditions, advancing understanding of geometric structures.

## Key findings

- Lipschitz graphs have non-dense direction sets.
- Non-Lipschitz graphs have dense but not complete direction sets.
- Non-graph sets determine all directions.

## Abstract

We consider the direction set determined by various subsets $E$ of Euclidean space and show that there is a trichotomy: Either (i) The subset is the graph of a Lipschitz function and the direction set is not dense in the sphere, (ii) The subset is the graph of a non-Lipschitz function and the direction set is dense but not everything, or (iii) The subset is not a graph (in a suitable sense) and every direction is determined by the set. We then explore a variety of results based on this trichotomy under additional assumptions on the set $E$.