A generalization of the concept of distance based on the simplex inequality

Original Paper
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Abstract

We introduce and discuss the concept of n-distance, a generalization to n elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality
$$\begin{aligned} d(x_1, \ldots , x_n)\le K\, \sum _{i=1}^n d(x_1, \ldots , x_n)_i^z, \quad x_1, \ldots , x_n, z \in X, \end{aligned}$$
where \(K=1\). Here \(d(x_1,\ldots ,x_n)_i^z\) is obtained from the function \(d(x_1,\ldots ,x_n)\) by setting its ith variable to z. We provide several examples of n-distances, and for each of them we investigate the infimum of the set of real numbers \(K\in \,]0,1]\) for which the inequality above holds. We also introduce a generalization of the concept of n-distance obtained by replacing in the simplex inequality the sum function with an arbitrary symmetric function.

Keywords

n-distance Simplex inequality Fermat point Smallest enclosing sphere 

Mathematics Subject Classification

Primary 39B72 Secondary 26D99 

Notes

Acknowledgements

This research is supported by the internal research project R-AGR-0500 of the University of Luxembourg.

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Copyright information

© The Managing Editors 2018

Authors and Affiliations

  1. 1.Mathematics Research UnitUniversity of LuxembourgEsch-sur-AlzetteLuxembourg

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