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Generalized-lush spaces revisited

  • V. Kadets
  • O. ZavarzinaEmail author
Original Paper

Abstract

We study geometric properties of GL-spaces. We demonstrate that every finite-dimensional GL-space is polyhedral; that in dimension 2 there are only two, up to isometry, GL-spaces, namely the space whose unit sphere is a square (like \(\ell _\infty ^2\) or \(\ell _1^2\)) and the space whose unit sphere is an equilateral hexagon. Finally, we characterise the spaces \(E = ({{\mathbb {R}}}^n, \Vert \cdot \Vert _E)\) with absolute norm such that for every collection \(X_1, \ldots , X_n\) of GL-spaces their E-sum is a GL-space.

Keywords

Tingley’s problem Mazur–Ulam property Polyhedral space GL-space Ultraproduct 

Mathematics Subject Classification

46B20 46B04 46B08 

Notes

Acknowledgements

We are grateful to the anonymous referee for valuable suggestions that improved the exposition. The research is done in frames of Ukrainian Ministry of Science and Education Research Program 0118U002036.

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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.School of Mathematics and InformaticsV.N. Karazin Kharkiv National UniversityKharkivUkraine

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