Asian Journal of Mathematics

Volume 25 (2021)

Number 4

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

Pages: 565 – 578



Paul Laurain (IMJ-Paris 7, Institut de Mathématiques de Jussieu, Equipe Géométrie et Dynamique, Paris, France)

Mircea Petrache (PUC Chile, Facultad de Matemáticas, Santiago, Chile)


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.


uniform measures, homogeneous measures, helices, higher codimension

2010 Mathematics Subject Classification

28C10, 49Q15, 51F20, 53A04

Received 5 May 2020

Accepted 5 January 2021

Published 25 April 2022