Characterising rectifiable metric spaces using tangent spaces

• 成果类型: Article
• 署名作者: Bate, David
• 署名单位: University of Warwick
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01136-7
• 发表日期: 2022
• 页码: 995-1070
• 关键词: n-rectifiability; sets; geometry; terms

摘要

We characterise rectifiable subsets of a complete metric space X in terms of local approximation, with respect to the Gromov-Hausdorff distance, by an n-dimensional Banach space. In fact, if E subset of X with H-n (E) < infinity and has positive lower density almost everywhere, we prove that it is sufficient that, at almost every point and each sufficiently small scale, E is approximated by a bi-Lipschitz image of Euclidean space. We also introduce a generalisation of Preiss's tangent measures that is suitable for the setting of arbitrary metric spaces and formulate our characterisation in terms of tangent measures. This definition is equivalent to that of Preiss when the ambient space is Euclidean, and equivalent to the measured Gromov-Hausdorff tangent space when the measure is doubling.