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### Smallest enclosing ball multidistance

I. Aguiló, J. Martín, G. Mayor, J. Suñer, O. Valero
2012 Communications in Information and Systems
Metric space, Fermat multidistance, smallest enclosing ball, midpoint property, Fermat property, m-dimensional Euclidian space. 1. Introduction.  ...  By using the diameter of the smallest enclosing ball of a set of points, we find conditions in order to ensure that the mentioned measure is a multidistance. Keywords.  ...  The Multidistance based on the Smallest Enclosing Ball. 3.1. The Smallest Enclosing Ball. Let (X, d) be a non trivial (|X| 2) proper metric space. Proposition 1.  ...

### Functionally Expressible Multidistances

Javier Martin, Gaspar Mayor, Oscar Valero
2011 Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011)
An example of non functionally expressible multidistance is exhibited.  ...  After introducing the concept of functionally expressible multidistance, several essential types of multidimensional aggregation functions are considered to construct such kind of multidistances.  ...  Figure 3 : 3 A smallest enclosing ball in the d ∞ -plane. D 1 (Figure 4 : 14 P 1 , . . . , P n ) A smallest enclosing ball in the d 1 -plane.  ...

### On the best constants associated with n-distances [article]

Gergely Kiss, Jean-Luc Marichal
2019 arXiv   pre-print
Finally, we discuss an interesting link between the concepts of n-distance and multidistance.  ...  It is easy to see that this is also the case for the diameter of the smallest enclosing Chebyshev ball in R q for any integer q ≥ 2, that is, d(x 1 , . . . , x n ) = max {i,j}⊆{1,...  ...  circle enclosing n points in R 2 (X = R 2 ). • Area of the smallest circle enclosing n points in R 2 (X = R 2 and n ≥ 3).  ...

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

Gergely Kiss, Jean-Luc Marichal, Bruno Teheux
2018 Contributions to Algebra and Geometry
Properties of multidistances as well as instances including the Fermat multidistance and smallest enclosing ball multidistances have been investigated for example in [2,    .  ...  In Section 4 we consider some geometric constructions (smallest enclosing sphere and number of directions) to define n-distances and study their corresponding best constants.  ...

### Data Fusion: Theory, Methods, and Applications [article]

Marek Gagolewski
2022 arXiv   pre-print
and A. 3 , 3 which relies on the CGAL library.Figure A.10 gives an exemplary Rcpp implementation of a routine to compute the smallest enclosing ball.  ...  The Euclidean 1-center (smallest enclosing ball radius) is given by: 1center d2 (x (1) , . . . , x (n) ) = arg min y∈R d n i=1 d 2 (x (i) , y), (2.8) where d 2 is the Euclidean metric. Example 2.9.  ...

### Data Fusion: Theory, Methods, and Applications [article]

Marek Gagolewski
2015 Zenodo
and A. 3 , 3 which relies on the CGAL library.Figure A.10 gives an exemplary Rcpp implementation of a routine to compute the smallest enclosing ball.  ...  The Euclidean 1-center (smallest enclosing ball radius) is given by: 1center d2 (x (1) , . . . , x (n) ) = arg min y∈R d n i=1 d 2 (x (i) , y), (2.8) where d 2 is the Euclidean metric. Example 2.9.  ...

### Medial/skeletal linking structures for multi-region configurations [article]

James Damon, Ellen Gasparovic
2015 arXiv   pre-print
Convex Hull: The smallest convex region which contains a configuration is the convex hull of the configuration.  ...  We view Ω as the region enclosed by the boundary B = ϕ(X), for ϕ : X → R n+1 a smooth embedding and X a smooth compact n-manifold.  ...  Then for the perturbation, we are considering the partial multijet of the multidistance functionρ i about the points (x There are two cases involving x (0) (= x (jp) 1 ), u (0) , and u (1) .  ...