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Hahn-Banach for metric functionals and horofunctions. (English) Zbl 1471.46019

The paper is concerned with notions and results which can be “transformed” from geometric Banach space theory to the purely metric space setting (see the survey in [A. Naor, in: Proceedings of the international congress of mathematicians, ICM 2018, Rio de Janeiro, Brazil, August 1–9, 2018. Volume I. Plenary lectures. Hackensack, NJ: World Scientific; Rio de Janeiro: Sociedade Brasileira de Matemática (SBM). 759–837 (2018; Zbl 1444.46019)]). In particular, it is proved that the Hahn-Banach theorem is valid for metric functionals but not for so-called horofunctions (see the survey in [A. Papadopoulos, Metric spaces, convexity and nonpositive curvature. Zürich: European Mathematical Society (EMS) (2014; Zbl 1296.53007)]). Some results on the existence of invariant metric functionals are proved. Moreover, it is shown that the metric (horofunction) boundary of every infinite Cayley graph contains at least two points.

MSC:

46B85 Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
54E35 Metric spaces, metrizability
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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References:

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