Abstract
C. Kimberling defined the concept of triangle center function in order to describe centers of triangles as points associated to some functions depending on the sidelengths, instead of in terms of geometrical properties. In this article we provide two definitions of n-gon center function for \(n\ge 3\): one of them in terms of the coordinates of the vertices and the other one by means of the lengths of the sides and the diagonals. Both of them are natural ways to generalize the concept of triangle center function, and we prove that they are equivalent. Moreover, we use n-gon center functions to associate to each polygon a point in the plane, that we call center. We also explore the problem of characterization of families of polygons in terms of these n-gon center functions and we study the relation between our new definitions and other approaches arising from Applied Mathematics.
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Acknowledgements
The second author is supported by a postdoctoral Grant (PEJD-2018-POST/TIC-9490) from UNED, co-financed by the Regional Government of Madrid with funds from the Youth Employment Initiative (YEI) of the European Union. The authors want to thank the editor for his helpful suggestions and comments.
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Prieto-Martínez, L.F., Sánchez-Cauce, R. Generalization of Kimberling’s Concept of Triangle Center for Other Polygons. Results Math 76, 81 (2021). https://doi.org/10.1007/s00025-021-01388-4
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DOI: https://doi.org/10.1007/s00025-021-01388-4