Stability and homotopy of a subset of the medial axis

Medial Axis is known to be unstable for non smooth objects. For an open set O, we define the Weak Feature Size, wfs, minimum distance between O^c and the critical points of the function distance to O^c. We introduce the "Lambda-Medial Axis" of O, Mλ, a subset of the Medial Axis of O which captures the Homotopy type of O when λ < wfs. We show that, at least for some "regular" values of λ, Mλ remains stable under Hausdorff distance perturbations of O^c.