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
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.
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This research is supported by the internal research project R-AGR-0500 of the University of Luxembourg.
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Kiss, G., Marichal, JL. & Teheux, B. A generalization of the concept of distance based on the simplex inequality. Beitr Algebra Geom 59, 247–266 (2018). https://doi.org/10.1007/s13366-018-0379-5
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DOI: https://doi.org/10.1007/s13366-018-0379-5