Compensated Convexity, Multiscale Medial Axis Maps and Sharp Regularity of the Squared Distance Function

We introduce a new stable mathematical model for locating and measuring the medial axis of geometric objects, called the quadratic multiscale medial axis map of scale $\lambda$, and prove a sharp regularity result for the squared-distance function to any closed non-empty subset $K$ of $\mathbb{R}^n$. Our results exploit properties of the function $C^l_{\lambda}(dist^2(\cdot;\, K))$ obtained by applying the quadratic lower compensated convex transform of parameter $\lambda$ to $dist^2(\cdot;\, K)$, the Euclidean squared-distance function to $K$. Using an estimate for the tight approximation of $dist^2(\cdot;\, K)$ by $C^l_{\lambda}(dist^2(\cdot;\, K))$, we prove $C^{1,1}$-regularity of $dist^2(\cdot;\, K)$ outside a neighbourhood of the closure of the medial axis $M_K$ of $K$, and give an asymptotic formula for $C^l_{\lambda}(dist^2(\cdot;\, K))$ in terms of the scaled squared distance to $K$ and to the convex hull of the set of points that realize the minimum distance to $K$. The multiscale medial axis map, $M_{\lambda}(\cdot;\, K)$, is a family of non-negative functions whose limit as $\lambda \to \infty$ exists and is called the multiscale medial axis landscape map, $M_{\infty}(\cdot;\, K)$. We show $M_{\infty}(\cdot;\, K)$ is strictly positive on the medial axis $M_K$ and zero elsewhere. We give conditions to ensure $M_{\lambda}(\cdot;\, K)$ keeps a constant height along parts of $M_K$ generated by two-point subsets with the height dependent on the distance between the generating points, so giving a hierarchy between different parts of $M_K$ that enables subsets of $M_K$ to be selected by thresholding. Given a compact subset $K$ of $\mathbb{R}^n$, while it is well known that $M_K$ is not Hausdorff stable, we prove $M_{\lambda}(\cdot;\, K)$ is stable under Hausdorff distance, and deduce implications for localization of the stable parts of $M_K$. Examples are included.