Higher dimensional Jordan curves

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 $\Gamma$ so that $E\subset \Gamma$ and $$\mathrm{diam}(E{)}^d+{\beta }^d(E)\approx {\mathcal{H}}^d(\Gamma ),$$ where ${\beta }^d$ 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 \textit{any} subset of Euclidean space.

arXiv comment76 pages. Thoroughly revised and extended version. Results unchanged. Title changed from `Sets with topology, the Analyst's TST, and applications'

References in corpus (2)

• A generalization of Reifenberg's theorem in $R^3$
• Necessary condition for rectifiability involving Wasserstein distance $W_2$

Cited by in corpus (4)

• A $d$-dimensional Analyst's Travelling Salesman Theorem for general sets in ${\mathbb{R}}^n$
• Cone and paraboloid points of arbitrary subsets of Euclidean space
• Analytic capacity and dimension of sets with plenty of big projections
• A note on the Hausdorff dimension of uniformly non flat sets with plenty of big projections