Abstract
Around 1930, Menger expressed his interest in the concept of abstract angle function. He introduced a general definition of this notion for metric and semi-metric spaces. He also proposed two problems concerning conformal embeddability of spaces endowed with an angle function into Euclidean spaces. These problems received attention in later years but only for some particular cases of metric spaces. In this article, we first update the definition of angle function to apply to the larger class of spaces with a notion of betweenness, which seem to us a more natural framework. In this new general setting, we solve the two problems proposed by Menger.
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Acknowledgements
The author want to thank the referees for their helpful suggestions and comments, which improved remarkably the first version of this paper. This work has been funded by Agencia Española de Investigación (Spain) through grant MTM2017-83740-P and Junta de Comunidades de Castilla-La Mancha (Spain) through grant SBPLY/19/18501/0000110.
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Prieto-Martínez, L.F. A Solution to Two Old Problems by Menger Concerning Angle Spaces. Mediterr. J. Math. 19, 194 (2022). https://doi.org/10.1007/s00009-022-02092-5
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DOI: https://doi.org/10.1007/s00009-022-02092-5
Keywords
- Angle space
- angle function
- conformal embeddability
- metric space
- isometrical embeddability
- Euclidean space