Dimension-independent quasiconvex connected-set conjecture

Let $d\geq 2$. For every subset $K\subset \mathbb{R}^d$, the main theorem asserts the existence of a connected set $\tilde\Gamma\subset\mathbb{R}^d$ containing $K$, with controlled one-dimensional Hausdorff measure and the quasiconvexity property that any two points can be joined within $\tilde\Gamma$ by a path whose length is at most a constant times their Euclidean distance. Dimension-independent quasiconvexity conjecture. The theorem should hold with constants independent of the dimension and, in fact, for subsets $K$ of an infinite-dimensional Hilbert space. The conjecture would extend the finite-dimensional construction of short quasiconvex supersets and connect it to the infinite-dimensional validity of the Traveling Salesman Theorem. The authors identify the control of the number of bridges in finite dimensions as the point of their proof that breaks down in infinite dimensions; no resolution is given here.