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

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.

Keywords

uniform measures, homogeneous measures, helices, higher codimension

2010 Mathematics Subject Classification

28C10, 49Q15, 51F20, 53A04

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Received 5 May 2020

Accepted 5 January 2021

Published 25 April 2022