| Reference : A generalization of the concept of distance based on the simplex inequality |
| Scientific journals : Article | |||
| Physical, chemical, mathematical & earth Sciences : Mathematics Engineering, computing & technology : Computer science | |||
| Computational Sciences | |||
| http://hdl.handle.net/10993/34025 | |||
| A generalization of the concept of distance based on the simplex inequality | |
| English | |
Kiss, Gergely  [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] | |
| Marichal, Jean-Luc [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] | |
| Teheux, Bruno [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] | |
| Jun-2018 | |
| Beitraege zur Algebra und Geometrie = Contributions to Algebra and Geometry | |
| Springer | |
| 59 | |
| 2 | |
| 247–266 | |
| Yes | |
| International | |
| 0138-4821 | |
| 2191-0383 | |
| [en] n-distance ; simplex inequality ; Fermat point ; smallest enclosing sphere | |
| [en] We introduce and discuss the concept of \emph{$n$-distance}, a generalization to $n$ elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality
\[ d(x_1, \ldots, x_n)~\leq~K\, \sum_{i=1}^n d(x_1, \ldots, x_n)_i^z{\,}, \qquad x_1, \ldots, x_n, z \in X, \] 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 $i$th 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 $K\in\left]0,1\right]$ 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. | |
| University of Luxembourg - UL | |
| R-AGR-0500 > MRO3 > 01/03/2015 - 28/02/2018 > MARICHAL Jean-Luc | |
| Researchers ; Professionals ; Students | |
| http://hdl.handle.net/10993/34025 | |
| https://arxiv.org/abs/1611.07826 |
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