The Inverse Weber Problem on the Plane and the Sphere
Weber’s inverse problem in the plane is to modify the positive weights associated with <i>n</i> fixed points in the plane at minimum cost, ensuring that a given point a priori becomes the Euclidean weighted geometric median. In this paper, we investigate Weber’s inverse problem in the pl...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/24/5000 |
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| author | Franco Rubio-López Obidio Rubio Rolando Urtecho Vidaurre |
| author_facet | Franco Rubio-López Obidio Rubio Rolando Urtecho Vidaurre |
| author_sort | Franco Rubio-López |
| collection | DOAJ |
| description | Weber’s inverse problem in the plane is to modify the positive weights associated with <i>n</i> fixed points in the plane at minimum cost, ensuring that a given point a priori becomes the Euclidean weighted geometric median. In this paper, we investigate Weber’s inverse problem in the plane and generalize it to the surface of the sphere. Our study uses a subspace orthogonal to a subspace generated by two vectors <i>X</i> and <i>Y</i> associated with the given points and weights. The main achievement of our work lies in determining a vector perpendicular to the vectors <i>X</i> and <i>Y</i>, in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>; which is used to determinate a solution of Weber’s inverse problem. In addition, lower bounds are obtained for the minimum of the Weber function, and an upper bound for the difference of the minimal of Weber’s direct and inverse problems. Examples of application at the plane and unit sphere are given. |
| first_indexed | 2024-03-08T20:34:14Z |
| format | Article |
| id | doaj.art-cf94141311484cce83d370595089ef27 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T20:34:14Z |
| publishDate | 2023-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-cf94141311484cce83d370595089ef272023-12-22T14:23:33ZengMDPI AGMathematics2227-73902023-12-011124500010.3390/math11245000The Inverse Weber Problem on the Plane and the SphereFranco Rubio-López0Obidio Rubio1Rolando Urtecho Vidaurre2Instituto de Investigación en Matemáticas, Departamento de Matemáticas, Universidad Nacional de Trujillo, Trujillo 13001, PeruInstituto de Investigación en Matemáticas, Departamento de Matemáticas, Universidad Nacional de Trujillo, Trujillo 13001, PeruInstituto de Investigación en Matemáticas, Departamento de Matemáticas, Universidad Nacional de Trujillo, Trujillo 13001, PeruWeber’s inverse problem in the plane is to modify the positive weights associated with <i>n</i> fixed points in the plane at minimum cost, ensuring that a given point a priori becomes the Euclidean weighted geometric median. In this paper, we investigate Weber’s inverse problem in the plane and generalize it to the surface of the sphere. Our study uses a subspace orthogonal to a subspace generated by two vectors <i>X</i> and <i>Y</i> associated with the given points and weights. The main achievement of our work lies in determining a vector perpendicular to the vectors <i>X</i> and <i>Y</i>, in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>; which is used to determinate a solution of Weber’s inverse problem. In addition, lower bounds are obtained for the minimum of the Weber function, and an upper bound for the difference of the minimal of Weber’s direct and inverse problems. Examples of application at the plane and unit sphere are given.https://www.mdpi.com/2227-7390/11/24/5000Weber’s problemWeber’s inverse problemlocation of servicesorthogonal space |
| spellingShingle | Franco Rubio-López Obidio Rubio Rolando Urtecho Vidaurre The Inverse Weber Problem on the Plane and the Sphere Mathematics Weber’s problem Weber’s inverse problem location of services orthogonal space |
| title | The Inverse Weber Problem on the Plane and the Sphere |
| title_full | The Inverse Weber Problem on the Plane and the Sphere |
| title_fullStr | The Inverse Weber Problem on the Plane and the Sphere |
| title_full_unstemmed | The Inverse Weber Problem on the Plane and the Sphere |
| title_short | The Inverse Weber Problem on the Plane and the Sphere |
| title_sort | inverse weber problem on the plane and the sphere |
| topic | Weber’s problem Weber’s inverse problem location of services orthogonal space |
| url | https://www.mdpi.com/2227-7390/11/24/5000 |
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