*Preprint*

**Inserted:** 1 jul 2021

**Last Updated:** 1 jul 2021

**Year:** 2021

**Abstract:**

We prove that the gradient of an anisotropic distance function $\delta_{K}^{\phi}$ from an arbitrary closed subset $K$ of $\mathbf{R}^{n}$ has a local Lipschitz property on the complement of the $\phi$-cut-locus of $K$, i.e., on the set $\mathbf{R}^{n} \sim (K \cup \textrm{Cut}^{\phi}(K))$. Next, we study the geometric properties of the set of points where $\delta^{\phi}$ is second-order pointwise differentiable. The anisotropy follows from working with a possibly non-Euclidean uniformly convex $\mathcal{C}^2$-norm $\phi$. The available known results allow only to conclude that the gradient of $\delta_{K}^{\phi}$ is locally Lipschitz on the interior of $\mathbf{R}^{n} \sim (K \cup \textrm{Cut}^{\phi}(K))$ which might be empty even if $K$ is a $\mathcal{C}^{1,\alpha}$ hypersurface or $K$ is the complementary of a convex body with $\mathcal{C}^{1,\alpha}$ boundary.