Midsets and Voronoi Type Decomposition with Respect to Closed Convex Sets

\def\R{\mathbb{R}} Let $\Omega_k$ denote the collection of all nonempty closed convex subsets of $\R^k$. We provide short proofs for the following: (i) $\{x\in \R^k:dist(x,A)=\varepsilon\}$ is a $C^1$-manifold of dimension $k-1$ for every $A\in \Omega_k\setminus \{\R^k\}$ and $\varepsilon>0$, (ii) $\{x\in \R^k:dist(x,A)=dist(x,B)\}$ is a $C^1$-manifold of dimension $k-1$ for any two disjoint $A, B\in \Omega_k$. We also study the distance of points in $\R^k$ to finitely many closed convex sets. Let $k,n\ge 2$ and $A=\bigcup_{j=1}^n A_j$, where $A_1,\ldots,A_n\in \Omega_k$ are pairwise disjoint. We consider a Voronoi type decomposition of $\R^k$ and establish some topological properties of its `conflict set'. Letting $X_p=\{x\in \R^k:|\{a\in A: \|x-a\| =dist(x,A)\}|=p\}$, we prove with the help of result (ii) stated above that $X_1\cup X_2$ is a connected dense open subset of $\R^k$ and that $\overline{X_2}=\bigcup_{p=2}^n X_p$.