# Classification of uniformly distributed measures of dimension $1$ in general codimension

## TL;DR

This paper classifies 1-dimensional uniform measures in Euclidean spaces, showing they are homogeneous when supported on connected sets, and provides partial results without this assumption, advancing understanding of measure geometry.

## Contribution

It offers a complete classification of uniform measures with connected support and partial results for the general case of 1-dimensional uniform measures in any dimension.

## Key findings

- Uniform measures with connected support are homogeneous.
- Partial classification of uniform measures without connected support.
- Advances the geometric understanding of 1-dimensional measures.

## Abstract

Starting with the work of Preiss on the geometry of measures, the classification of uniform measures in $\mathbb R^d$ has remained open, except for $d=1$ and for compactly supported measures in $d=2$, and for codimension $1$. In this paper we study $1$-dimensional measures in $\mathbb R^d$ for all $d$ and classify uniform measures with connected $1$-dimensional support, which turn out to be homogeneous measures. We provide as well a partial classification of general uniform measures of dimension $1$ in the absence of the connected support hypothesis.