The Voronoi inverse mapping

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• 作者：M.A. Goberna^a ; ^{mgoberna@ua.es" class="auth_mail" title="E-mail the corresponding author} ; J.E. Martí ; nez-Legaz^b ; ^{JuanEnrique.Martinez.Legaz@uab.cat" class="auth_mail" title="E-mail the corresponding author} ; V.N. Vera de Serio^c ; ^{virginia.vera@fce.uncu.edu.ar" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A39 ; 51M20 ; 52C22
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 September 2016
• 年：2016
• 卷：504
• 期：Complete
• 页码：248-271
• 全文大小：469 K

文摘

Given an arbitrary set T in the Euclidean space whose elements are called sites, and a particular site s, the Voronoi cell of s  , denoted by V_T(s)$V_T\left(s\right)$, consists of all points closer to s than to any other site. The Voronoi mapping of s  , denoted by c26a8ea1d2fe392475e979d91990" title="Click to view the MathML source">ψ_s${\psi }_s$, associates to each set T∋s$T\ni s$ the Voronoi cell V_T(s)$V_T\left(s\right)$ of s w.r.t. T  . These Voronoi cells are solution sets of linear inequality systems, so they are closed convex sets. In this paper we study the Voronoi inverse problem consisting in computing, for a given closed convex set F∋s$F\ni s$, the family of sets T∋s$T\ni s$ such that ψ_s(T)=F${\psi }_s\left(T\right)=F$. More in detail, the paper analyzes relationships between the elements of this family, ${\psi }_s^{-1}\left(F\right)$, and the linear representations of F  , provides explicit formulas for maximal and minimal elements of ${\psi }_s^{-1}\left(F\right)$, and studies the closure operator that assigns, to each closed set T containing s  , the largest element of ${\psi }_s^{-1}\left(F\right)$, where F=V_T(s)$F=V_T\left(s\right)$.