The difference vectors for convex sets and a resolution of the geometry conjecture
The geometry conjecture, which was posed nearly a quarter of a century ago, states that the fixed point set of the composition of projectors onto nonempty closed convex sets in Hilbert space is actually equal to the intersection of certain translations of the underlying sets.In this paper, we provid...
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| Format: | Article |
| Language: | English |
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Université de Montpellier
2021-07-01
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| Series: | Open Journal of Mathematical Optimization |
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| Online Access: | https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.7/ |
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| _version_ | 1825204958930141184 |
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| author | Alwadani, Salihah Bauschke, Heinz H. Revalski, Julian P. Wang, Xianfu |
| author_facet | Alwadani, Salihah Bauschke, Heinz H. Revalski, Julian P. Wang, Xianfu |
| author_sort | Alwadani, Salihah |
| collection | DOAJ |
| description | The geometry conjecture, which was posed nearly a quarter of a century ago, states that the fixed point set of the composition of projectors onto nonempty closed convex sets in Hilbert space is actually equal to the intersection of certain translations of the underlying sets.In this paper, we provide a complete resolution of the geometry conjecture. Our proof relies on monotone operator theory. We revisit previously known results and provide various illustrative examples. Comments on the numerical computation of the quantities involved are also presented. |
| format | Article |
| id | doaj-art-460c6d8a1e804d118cd646709e992c39 |
| institution | Kabale University |
| issn | 2777-5860 |
| language | English |
| publishDate | 2021-07-01 |
| publisher | Université de Montpellier |
| record_format | Article |
| series | Open Journal of Mathematical Optimization |
| spelling | doaj-art-460c6d8a1e804d118cd646709e992c392025-02-07T14:02:30ZengUniversité de MontpellierOpen Journal of Mathematical Optimization2777-58602021-07-01211810.5802/ojmo.710.5802/ojmo.7The difference vectors for convex sets and a resolution of the geometry conjectureAlwadani, Salihah0Bauschke, Heinz H.1Revalski, Julian P.2Wang, Xianfu3Mathematics University of British Columbia Kelowna, B.C. V1V 1V7 CanadaMathematics University of British Columbia Kelowna, B.C. V1V 1V7 CanadaInstitute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev str., Block 8 1113 Sofia BulgariaMathematics University of British Columbia Kelowna, B.C. V1V 1V7 CanadaThe geometry conjecture, which was posed nearly a quarter of a century ago, states that the fixed point set of the composition of projectors onto nonempty closed convex sets in Hilbert space is actually equal to the intersection of certain translations of the underlying sets.In this paper, we provide a complete resolution of the geometry conjecture. Our proof relies on monotone operator theory. We revisit previously known results and provide various illustrative examples. Comments on the numerical computation of the quantities involved are also presented.https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.7/Attouch–Théra dualitycircular right shift operatorconvex setscyclefixed point setmonotone operator theoryprojectors. |
| spellingShingle | Alwadani, Salihah Bauschke, Heinz H. Revalski, Julian P. Wang, Xianfu The difference vectors for convex sets and a resolution of the geometry conjecture Open Journal of Mathematical Optimization Attouch–Théra duality circular right shift operator convex sets cycle fixed point set monotone operator theory projectors. |
| title | The difference vectors for convex sets and a resolution of the geometry conjecture |
| title_full | The difference vectors for convex sets and a resolution of the geometry conjecture |
| title_fullStr | The difference vectors for convex sets and a resolution of the geometry conjecture |
| title_full_unstemmed | The difference vectors for convex sets and a resolution of the geometry conjecture |
| title_short | The difference vectors for convex sets and a resolution of the geometry conjecture |
| title_sort | difference vectors for convex sets and a resolution of the geometry conjecture |
| topic | Attouch–Théra duality circular right shift operator convex sets cycle fixed point set monotone operator theory projectors. |
| url | https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.7/ |
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