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Journal Article

Classification of uniformly distributed measures of dimension 1 in general codimension

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Petrache,  Mircea
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Laurain, P., & Petrache, M. (2021). Classification of uniformly distributed measures of dimension 1 in general codimension. Asian Journal of Mathematics, 25(4), 565-578. doi:10.4310/AJM.2021.v25.n4.a6.


Cite as: https://hdl.handle.net/21.11116/0000-000A-7DF5-6
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.