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Journal of Convex Analysis 25 (2018), No. 4, 1345--1354
Copyright Heldermann Verlag 2018

Midsets and Voronoi Type Decomposition with Respect to Closed Convex Sets

T. K. Subrahmonian Moothathu
School of Mathematics and Statistics, University of Hyderabad, Hyderabad 500 046, India


\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 $\overline{X_2}=\bigcup_{p=2}^n X_p$.

Keywords: Euclidean geometry, closed convex sets, Voronoi decomposition, midsets.

MSC: 52A20

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