Reference : A generalization of the concept of distance based on the simplex inequality
 Document type : Scientific journals : Article Discipline(s) : Physical, chemical, mathematical & earth Sciences : MathematicsEngineering, computing & technology : Computer science Focus Areas : Computational Sciences To cite this reference: http://hdl.handle.net/10993/34025
 Title : A generalization of the concept of distance based on the simplex inequality Language : English Author, co-author : 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 >] Publication date : Jun-2018 Journal title : Beitraege zur Algebra und Geometrie = Contributions to Algebra and Geometry Publisher : Springer Volume : 59 Issue/season : 2 Pages : 247–266 Peer reviewed : Yes Audience : International ISSN : 0138-4821 e-ISSN : 2191-0383 Keywords : [en] n-distance ; simplex inequality ; Fermat point ; smallest enclosing sphere Abstract : [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. Funders : University of Luxembourg - UL Name of the research project : R-AGR-0500 > MRO3 > 01/03/2015 - 28/02/2018 > MARICHAL Jean-Luc Target : Researchers ; Professionals ; Students Permalink : http://hdl.handle.net/10993/34025 Other URL : https://arxiv.org/abs/1611.07826

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