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We introduce a new stable mathematical model for locating and measuring themedial axis of geometric objects, called the quadratic multiscale medial axismap of scale $\lambda$, and prove a sharp regularity result for thesquared-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 ofparameter $\lambda$ to $dist^2(\cdot;\, K)$, the Euclidean squared-distancefunction 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 theclosure 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 minimumdistance to $K$. The multiscale medial axis map, $M_{\lambda}(\cdot;\, K)$, isa family of non-negative functions whose limit as $\lambda \to \infty$ existsand 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 subsetswith the height dependent on the distance between the generating points, sogiving 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, weprove $M_{\lambda}(\cdot;\, K)$ is stable under Hausdorff distance, and deduceimplications for localization of the stable parts of $M_K$. Examples areincluded.

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