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Nonstandard n-distances based on certain geometric constructions

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Abstract

The concept of n-distance was recently introduced to generalize the classical definition of distance to functions of n arguments. In this paper we investigate this concept through a number of examples based on certain geometrical constructions. In particular, our study shows to which extent the computation of the best constant associated with an n-distance may sometimes be difficult and tricky. It also reveals that two important graph theoretical concepts, namely the total length of the Euclidean Steiner tree and the total length of the minimal spanning tree constructed on n points, are instances of n-distances.

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Notes

  1. We consider acute triangles only to avoid highly discontinuous situations (e.g., when considering the three points \(x_1=(-1,0)\), \(x_2=(0,0)\), \(x_3=(1,\varepsilon )\) in \({\mathbb {R}}^2\) for a small value of \(\varepsilon \)).

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Acknowledgements

G. Kiss is supported by Premium Postdoctoral Fellowship of the Hungarian Academy of Sciences and by the Hungarian National Foundation for Scientific Research, Grant No. K124749.

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Authors and Affiliations

  1. Alfred Renyi Institute of Mathematics, Hungarian Academy of Science, Reáltanoda u. 13-15, 1053, Budapest, Hungary

    Gergely Kiss

  2. Mathematics Research Unit, University of Luxembourg, Maison du Nombre, 6, avenue de la Fonte, 4364, Esch-sur-Alzette, Luxembourg

    Jean-Luc Marichal

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  1. Gergely Kiss
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  2. Jean-Luc Marichal
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Correspondence to Jean-Luc Marichal.

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Kiss, G., Marichal, JL. Nonstandard n-distances based on certain geometric constructions. Beitr Algebra Geom 64, 107–126 (2023). https://doi.org/10.1007/s13366-022-00623-5

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  • DOI: https://doi.org/10.1007/s13366-022-00623-5

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