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Centre of a compact convex set

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Abstract

We introduce the notion centre of a convex set and study the space of continuous affine functions on a compact convex set with a centre. We show that these spaces are precisely the dual of a base normed space in which the underlying base has a (unique) centre. We also characterize the corresponding base norm space. We obtain a condition on a compact, balanced, convex subset of a locally convex space, so that the corresponding space of continuous affine functions on the convex set is an absolute order unit space. Similarly, we characterize a condition on the base with a centre of a base normed space, so that the latter becomes an absolutely base normed space.

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Acknowledgements

The author thanks the referee for his suggestions. This research was partially supported by Science and Engineering Research Board, Department of Science and Technology, Government of India sponsored Mathematical Research Impact Centric Support project (Reference no. MTR/2020/000017).

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Authors and Affiliations

  1. School of Mathematical Sciences, National Institute of Science Education and Research Bhubaneswar, An OCC of Homi Bhabha National Institute, P.O. Jatni, Khurda, Odisha, 752050, India

    Anil Kumar Karn

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  1. Anil Kumar Karn
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Correspondence to Anil Kumar Karn.

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Communicated by Dirk Werner.

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Karn, A.K. Centre of a compact convex set. Banach J. Math. Anal. 16, 68 (2022). https://doi.org/10.1007/s43037-022-00222-5

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  • DOI: https://doi.org/10.1007/s43037-022-00222-5

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