Higher dimensional Jordan curves

2019/08/27 by Michele Villa, Villa, Michele · 1 citation
Computer Science · Mathematics · #28A12 #28A75 #28A78 #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1908.10289

openalex publication_date 2019/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We address the question of what is the correct higher dimensional analogue of Jordan curves from the point of view of quantitative rectifiability. More precisely, we show that 'topologically stable' sets can be used as covering objects in Analyst's Travelling Salesman Theorem-type theorems: if E is lower d-regular (in a certain suitable sense), then we show that there exists a topologically stable surface Γ so that E ⊂ Γ and diam(E)^d + β^d(E) ≈ H^d(Γ), where β^d is a term quantifying the curvature of E. A corollary of the main result of this paper and a construction by Hyde, is a higher dimensional analogue of Peter Jones TST, valid for any subset of Euclidean space.

Citations

• Solution of the Plateau problem for m-dimensional surfaces of varying topological type

Cited by

• A d-dimensional Analyst's Travelling Salesman Theorem for general sets in ℝⁿ

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