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A note on symmetric separation in Banach spaces

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Abstract

We present some new results on the symmetric Kottman constant \(K^s(X)\) of a Banach space X and its relationship with the Kottman constant. We show that \(K^s(X)>1\), for every infinite-dimensional Banach space, thereby solving a problem by Castillo and Papini. We also investigate such constant in the class of Banach spaces admitting \(c_0\) spreading models, answering in particular one question from our previous joint paper with Hájek and Kania.

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Acknowledgements

Part of the results in this paper originates from a conversation with Pavlos Motakis, at the conference Non Linear Functional Analysis held at CIRM, Marseille, in March 2018. In particular, Pavlos Motakis suggested us the second proof of Theorem A presented in Sect. 3 and conjectured that some form of Theorem 4.4 might be true. We are most grateful to Pavlos for sharing his insight with us and for his interest in the topic. Moreover, we wish to thank Pier Luigi Papini for carefully reading our note and for his many suggestions which helped to improve the clarity of the paper. Finally, we thank the anonymous referee for the helpful report.

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  1. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27, Praha 6, Czech Republic

    Tommaso Russo

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  1. Tommaso Russo
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Correspondence to Tommaso Russo.

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Research of the author was supported by the project International Mobility of Researchers in CTU CZ.02.2.69/0.0/0.0/16\(\_\)027/0008465 and by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM), Italy.

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Russo, T. A note on symmetric separation in Banach spaces. RACSAM 113, 3649–3658 (2019). https://doi.org/10.1007/s13398-019-00722-4

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