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We construct a Lipschitz function on ${ℝ}^2$ which is locally convex on the complement of some totally disconnected compact set but not convex. Existence of such function disproves a theorem that appeared in a paper by L. Pasqualini and was also cited by other authors.

We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets $A\subset {ℝ}^2$. We prove that, for such $A$, the distance function $d_A=\mathrm{dist}(·,A)$ is a “DC aura” for $A$, which implies that each closed locally WDC set in ${ℝ}^2$ is a WDC set. Another consequence is that compact WDC subsets of ${ℝ}^2$ form a Borel subset of the space of all compact sets.

We give a complete characterization of closed sets $F\subset {ℝ}^2$ whose distance function $d_F:=\mathrm{dist}(·,F)$ is DC (i.e., is the difference of two convex functions on ${ℝ}^2$). Using this characterization, a number of properties of such sets is proved.

We deal with the so-called Ahlfors regular sets (also known as $s$-regular sets) in metric spaces. First we show that those sets correspond to a certain class of tree-like structures. Building on this observation we then study the following question: Under which conditions does the limit ${lim}_{\epsilon \to 0+}{\epsilon }^{s}N(\epsilon ,K)$ exist, where $K$ is an $s$-regular set and $N(\epsilon ,K)$ is for instance the $\epsilon$-packing number of $K$?