Dual medial axes in finite-dimensional normed spaces

We study medial axes of nonempty closed sets in finite-dimensional real normed spaces whose norms are $C^1$ away from the origin. Using the differential of the norm, we define the dual $k$-medial axis by requiring the contact covectors associated with nearest points to have affine dimension at least $k-1$. In dimension $n$, we prove that this set is $(n-k+1)$-rectifiable for every $1\le k\le n+1$, and hence has Hausdorff dimension at most $n-k+1$. We give a direct geometric proof and characterize the dual medial axes through the superdifferential of the squared distance function, obtaining an alternative proof from a theorem of Alberti, Ambrosio, and Cannarsa on singularities of semiconvex functions. Under the additional assumption of strict convexity, the ordinary and dual medial axes coincide for $k=2,3$, yielding the corresponding rectifiability results for ordinary medial axes. For $k=4$, however, the ordinary rectifiability conclusion can fail even for norms arbitrarily close to Euclidean: we construct norms on ${\mathbb{R}}^3$ with smooth positively curved unit spheres arbitrarily $C^{\mathrm{\infty }}$-close to the Euclidean sphere, together with four-point sets whose ordinary $4$-medial axes contain nondegenerate line segments.