A non-geodesic analogue of Reshetnyak’s majorization theorem
For any real number κ\kappa and any integer n≥4n\ge 4, the Cycln(κ){{\rm{Cycl}}}_{n}\left(\kappa ) condition introduced by Gromov (CAT(κ)-spaces: construction and concentration, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), (Geom. i Topol. 7), 100–140, 299–300) is a...
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| Format: | Article |
| Language: | English |
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De Gruyter
2023-03-01
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| Series: | Analysis and Geometry in Metric Spaces |
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| Online Access: | https://doi.org/10.1515/agms-2022-0151 |
| _version_ | 1797848777300639744 |
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| author | Toyoda Tetsu |
| author_facet | Toyoda Tetsu |
| author_sort | Toyoda Tetsu |
| collection | DOAJ |
| description | For any real number κ\kappa and any integer n≥4n\ge 4, the Cycln(κ){{\rm{Cycl}}}_{n}\left(\kappa ) condition introduced by Gromov (CAT(κ)-spaces: construction and concentration, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), (Geom. i Topol. 7), 100–140, 299–300) is a necessary condition for a metric space to admit an isometric embedding into a CAT(κ){\rm{CAT}}\left(\kappa ) space. For geodesic metric spaces, satisfying the Cycl4(κ){{\rm{Cycl}}}_{4}\left(\kappa ) condition is equivalent to being CAT(κ){\rm{CAT}}\left(\kappa ). In this article, we prove an analogue of Reshetnyak’s majorization theorem for (possibly non-geodesic) metric spaces that satisfy the Cycl4(κ){{\rm{Cycl}}}_{4}\left(\kappa ) condition. It follows from our result that for general metric spaces, the Cycl4(κ){{\rm{Cycl}}}_{4}\left(\kappa ) condition implies the Cycln(κ){{\rm{Cycl}}}_{n}\left(\kappa ) conditions for all integers n≥5n\ge 5. |
| first_indexed | 2024-04-09T18:32:51Z |
| format | Article |
| id | doaj.art-3d842d290e644eb3986e4a571148ed5c |
| institution | Directory Open Access Journal |
| issn | 2299-3274 |
| language | English |
| last_indexed | 2024-04-09T18:32:51Z |
| publishDate | 2023-03-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Analysis and Geometry in Metric Spaces |
| spelling | doaj.art-3d842d290e644eb3986e4a571148ed5c2023-04-11T17:07:12ZengDe GruyterAnalysis and Geometry in Metric Spaces2299-32742023-03-0111165770010.1515/agms-2022-0151A non-geodesic analogue of Reshetnyak’s majorization theoremToyoda Tetsu0Kogakuin University, 2665-1, Nakano, Hachioji, Tokyo, 192-0015 JapanFor any real number κ\kappa and any integer n≥4n\ge 4, the Cycln(κ){{\rm{Cycl}}}_{n}\left(\kappa ) condition introduced by Gromov (CAT(κ)-spaces: construction and concentration, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), (Geom. i Topol. 7), 100–140, 299–300) is a necessary condition for a metric space to admit an isometric embedding into a CAT(κ){\rm{CAT}}\left(\kappa ) space. For geodesic metric spaces, satisfying the Cycl4(κ){{\rm{Cycl}}}_{4}\left(\kappa ) condition is equivalent to being CAT(κ){\rm{CAT}}\left(\kappa ). In this article, we prove an analogue of Reshetnyak’s majorization theorem for (possibly non-geodesic) metric spaces that satisfy the Cycl4(κ){{\rm{Cycl}}}_{4}\left(\kappa ) condition. It follows from our result that for general metric spaces, the Cycl4(κ){{\rm{Cycl}}}_{4}\left(\kappa ) condition implies the Cycln(κ){{\rm{Cycl}}}_{n}\left(\kappa ) conditions for all integers n≥5n\ge 5.https://doi.org/10.1515/agms-2022-0151reshetnyak’s majorization theoremcat(κ) spacecycln(κ) space⊠-inequalitiesweighted quadruple inequalitiesprimary 53c23secondary 51f99 |
| spellingShingle | Toyoda Tetsu A non-geodesic analogue of Reshetnyak’s majorization theorem Analysis and Geometry in Metric Spaces reshetnyak’s majorization theorem cat(κ) space cycln(κ) space ⊠-inequalities weighted quadruple inequalities primary 53c23 secondary 51f99 |
| title | A non-geodesic analogue of Reshetnyak’s majorization theorem |
| title_full | A non-geodesic analogue of Reshetnyak’s majorization theorem |
| title_fullStr | A non-geodesic analogue of Reshetnyak’s majorization theorem |
| title_full_unstemmed | A non-geodesic analogue of Reshetnyak’s majorization theorem |
| title_short | A non-geodesic analogue of Reshetnyak’s majorization theorem |
| title_sort | non geodesic analogue of reshetnyak s majorization theorem |
| topic | reshetnyak’s majorization theorem cat(κ) space cycln(κ) space ⊠-inequalities weighted quadruple inequalities primary 53c23 secondary 51f99 |
| url | https://doi.org/10.1515/agms-2022-0151 |
| work_keys_str_mv | AT toyodatetsu anongeodesicanalogueofreshetnyaksmajorizationtheorem AT toyodatetsu nongeodesicanalogueofreshetnyaksmajorizationtheorem |