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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300891&_mathId=si1.gif&_user=111111111&_pii=S0024379516300891&_rdoc=1&_issn=00243795&md5=c1fa4e3abd58e57aec2d1b65b486384f" title="Click to view the MathML source">V_T(s)class="mathContainer hidden">class="mathCode">$V_T\left(s\right)$, consists of all points closer to s than to any other site. The Voronoi mapping of s  , denoted by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300891&_mathId=si2.gif&_user=111111111&_pii=S0024379516300891&_rdoc=1&_issn=00243795&md5=2701c26a8ea1d2fe392475e979d91990" title="Click to view the MathML source">ψ_sclass="mathContainer hidden">class="mathCode">${\psi }_s$, associates to each set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300891&_mathId=si3.gif&_user=111111111&_pii=S0024379516300891&_rdoc=1&_issn=00243795&md5=35f266e0625b1c3c5e2217fb1416cdbb" title="Click to view the MathML source">T∋sclass="mathContainer hidden">class="mathCode">$T\ni s$ the Voronoi cell class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300891&_mathId=si1.gif&_user=111111111&_pii=S0024379516300891&_rdoc=1&_issn=00243795&md5=c1fa4e3abd58e57aec2d1b65b486384f" title="Click to view the MathML source">V_T(s)class="mathContainer hidden">class="mathCode">$V_T\left(s\right)$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300891&_mathId=si5.gif&_user=111111111&_pii=S0024379516300891&_rdoc=1&_issn=00243795&md5=3f2d7321cc63c0d3d3dbae36282152ac" title="Click to view the MathML source">F∋sclass="mathContainer hidden">class="mathCode">$F\ni s$, the family of sets class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300891&_mathId=si3.gif&_user=111111111&_pii=S0024379516300891&_rdoc=1&_issn=00243795&md5=35f266e0625b1c3c5e2217fb1416cdbb" title="Click to view the MathML source">T∋sclass="mathContainer hidden">class="mathCode">$T\ni s$ such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300891&_mathId=si6.gif&_user=111111111&_pii=S0024379516300891&_rdoc=1&_issn=00243795&md5=12d515e10e2b47c4a31dca81bb2f5738" title="Click to view the MathML source">ψ_s(T)=Fclass="mathContainer hidden">class="mathCode">${\psi }_s\left(T\right)=F$. More in detail, the paper analyzes relationships between the elements of this family, class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300891&_mathId=si342.gif&_user=111111111&_pii=S0024379516300891&_rdoc=1&_issn=00243795&md5=dd41572013a959c17e3366240021040c">class="imgLazyJSB inlineImage" height="18" width="57" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300891-si342.gif">class="mathContainer hidden">class="mathCode">${\psi }_s^{-1}\left(F\right)$, and the linear representations of F  , provides explicit formulas for maximal and minimal elements of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300891&_mathId=si342.gif&_user=111111111&_pii=S0024379516300891&_rdoc=1&_issn=00243795&md5=dd41572013a959c17e3366240021040c">class="imgLazyJSB inlineImage" height="18" width="57" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300891-si342.gif">class="mathContainer hidden">class="mathCode">${\psi }_s^{-1}\left(F\right)$, and studies the closure operator that assigns, to each closed set T containing s  , the largest element of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300891&_mathId=si342.gif&_user=111111111&_pii=S0024379516300891&_rdoc=1&_issn=00243795&md5=dd41572013a959c17e3366240021040c">class="imgLazyJSB inlineImage" height="18" width="57" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300891-si342.gif">class="mathContainer hidden">class="mathCode">${\psi }_s^{-1}\left(F\right)$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300891&_mathId=si9.gif&_user=111111111&_pii=S0024379516300891&_rdoc=1&_issn=00243795&md5=8e70a73017a44b84d616489f378e0ea1" title="Click to view the MathML source">F=V_T(s)class="mathContainer hidden">class="mathCode">$F=V_T\left(s\right)$.