Higher dimensional Jordan curves

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 $\mathrm{\Gamma }$ so that $E\subset \mathrm{\Gamma }$ and $$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(E{)}^d+{\beta }^d(E)\approx {\mathcal{H}}^d(\mathrm{\Gamma }),$$ 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 $\mathit{\text{any}}$ subset of Euclidean space.