How to Stabilize the Medial Axis Transform

We've had a draft paper on github for a while related to our study of Blum's medial axis transform $M(A)$ of open subsets $A \subset {\bf{R}}^n$. We were seeking an elementary proof of the homotopy-isomorphism between $M(A)$ and $A$, but there were some technical difficulties related to an interesting paper by Nirenberg-Li. Here technical hypotheses on the boundary of $A$ are nececessary. In general, the boundary of an open subset $A$ of ${\bf{R}}^n$ satisfies $$dim_} (\partial A) \geq dim(A) -1.$$ This is because there exists continuous surjection of $\partial{A}$ onto $n-1$ hyperplanes in ${\bf{R}}^n$.

In this post we want to propose a solution to the notorious instability of $M(A)$ with respect to perturbations $A'$ of $A$. In practice, what happens is that small perturbations or "bubbles" $A'$ will cause large branches to appear in $M(A')$. This is a cause of discontinuity with respect to Gromov-Hausdorff distance of the medial axes $M(A)$.

[The following figure is taken from a paper on MAT transform, we don't recall which one. Apologies!]

However, what our Reduction-to-Singularity method shows [link to thesis] is that these tendrils are homotopically trivial, and therefore the maximal reduction of $M(A)$ is continuous with respect to GH-topology.

More formally our thesis establishes a maximal reduction $M(A) \leadsto W(A)$, where $W(A) \hookrightarrow M(A)$ is a homotopy-isomorphism. We are proposing that $W(A)$ is basically $M(A)$ minus the trivial branches. And what we propose is that $W(A)$ is continuous with respect to GH-topology. This means if $A_k$, $k=1,2,3, \ldots$ is a sequence of open subsets converging in GH-topology $$\lim_{k\to +\infty} A_k = A_\infty, $$ then $W(A_k)$ GH-converges to $W(A_\infty)$. By contrast we cannot expect any convergence between $M(A_k)$ and $M(A_\infty)$ because $M$ is generally only upper semicontinuous.

Warning. The above paragraphs are speculative at this stage, and not formally "proven".