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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