Abstract

Let Ωk\Omega_k denote the collection of all nonempty closed convex subsets of Rk\R^k. We provide short proofs for the following: (i) {x∈Rk:dist(x,A)=ε}\{x\in \R^k:dist(x,A)=\varepsilon\} is a C1C^1-manifold of dimension k−1k-1 for every A∈Ωk∖{Rk}A\in \Omega_k\setminus \{\R^k\} and ε>0\varepsilon>0, (ii) {x∈Rk:dist(x,A)=dist(x,B)}\{x\in \R^k:dist(x,A)=dist(x,B)\} is a C1C^1-manifold of dimension k−1k-1 for any two disjoint A,B∈ΩkA, B\in \Omega_k. We also study the distance of points in Rk\R^k to finitely many closed convex sets. Let k,n≥2k,n\ge 2 and A=⋃j=1nAjA=\bigcup_{j=1}^n A_j, where A1,…,An∈ΩkA_1,\ldots,A_n\in \Omega_k are pairwise disjoint. We consider a Voronoi type decomposition of Rk\R^k and establish some topological properties of its `conflict set'. Letting Xp={x∈Rk:∣{a∈A:∥x−a∥=dist(x,A)}∣=p}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 X1∪X2X_1\cup X_2 is a connected dense open subset of Rk\R^k and that X2‾=⋃p=2nXp\overline{X_2}=\bigcup_{p=2}^n X_p.

Author information

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T. K. Subrahmonian Moothathu

School of Mathematics and Statistics, University of Hyderabad, Hyderabad 500 046, India

tksubru@gmail.com

Keywords

  • Euclidean geometry
  • closed convex sets
  • Voronoi decomposition
  • midsets

Mathematics Subject Classification

52A20

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Suggested citation

T. K. Subrahmonian Moothathu. “Midsets and Voronoi Type Decomposition with Respect to Closed Convex Sets.” Journal of Convex Analysis 25 (2018), No. 4, 1345–1354. https://doi.org/10.68381/jca25082

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press LLC.