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arXiv · 1905.09601

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

Paul Laurain·Mircea Petrache
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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.

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Paul Laurain, Mircea Petrache. 2019-05-23. Classification of uniformly distributed measures of dimension $1$ in general codimension. https://arxiv.org/abs/1905.09601

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