[en] 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.
Disciplines :
Mathématiques
Auteur, co-auteur :
Kiss, Gergely; Alfred Renyi Institute of Mathematics, Hungarian Academy of Science
MARICHAL, Jean-Luc ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
Co-auteurs externes :
yes
Langue du document :
Anglais
Titre :
Nonstandard n-distances based on certain geometric constructions
Boltyanski, V., Martini, H., Soltan, V. Geometric Methods and Optimization Problems. Combinatorial Optimization, 4. Springer, Boston, MA, 1999
Brazil, M., Graham, R.L., Thomas, D.A., Zachariasen, M.: On the history of the Euclidean Steiner tree problem. Arch. Hist. Exact Sci. 68(3), 327–354 (2014) DOI: 10.1007/s00407-013-0127-z
Cieslik, D.: Shortest Connectivity. An Introduction with Applications in Phylogeny. Combinatorial Optimization, 17. Springer-Verlag, New York, 2005
Deza, M.M., Deza, E.: Encyclopedia of Distances, third edition. Springer, 2014
Dhage, B.C.: Generalised metric spaces and mappings with fixed point. Bull. Calcutta Math. Soc. 84, 329–336 (1992)
Kiss, G., Marichal, J.-L.: On the best constants associated with n -distances. Acta Math. Hungar. 161(1), 341–365 (2020) DOI: 10.1007/s10474-020-01023-8
Kiss, G., Marichal, J.-L., Teheux, B.: An extension of the concept of distance as functions of several variables. Proc. 36th Linz Seminar on Fuzzy Set Theory (LINZ 2016), Linz, Austria, Feb. 2-6, pp. 53–56, 2016
Kiss, G., Marichal, J.-L., Teheux, B.: A generalization of the concept of distance based on the simplex inequality. Beitr. Algebra Geom. 59(2), 247–266 (2018) DOI: 10.1007/s13366-018-0379-5
Wu, B.Y., Chao, K.-M.: Spanning Trees and Optimization Problems. Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton (2004)
