Bi-Lipschitz charts at regular points of non-collapsed RCD spaces

Establish whether every regular point of a non-collapsed RCD space admits a bi-Lipschitz chart.

Background

The paper discusses the general problem of promoting infinitesimal Euclidean structure at regular points of RCD spaces to genuine local metric charts. In the non-collapsed setting, Reifenberg-type results provide such charts near almost regular points, but the authors note that a broader conjecture remains unresolved. The positive injectivity-radius theorem proved in the paper gives bi-Lipschitz charts under an additional global metric hypothesis, and therefore does not settle the conjecture for arbitrary regular points.

The problem concerns the existence of quantitative local coordinates in the non-collapsed RCD category, independently of the stronger regularity questions addressed under positive injectivity radius.