An n-ary generalization of the concept of distance
Gergely Kiss, Jean-Luc Marichal, and Bruno Teheux
Mathematics Research Unit, University of Luxembourg
gergely.kiss@uni.lu, jean-luc.marichal@uni.lu, bruno.teheux@uni.lu
Generalizations of the concept of distance in which n ≥ 3 elements are con- sidered have been investigated by several authors (see [1, Chapter 3] and the references therein). The general idea is to provide some functions that measure a degree of dispersion among n points. In this talk, we consider the class of n-distances, which are defined as follows.
Definition 1. Let X be a nonempty set and n ≥ 2. A map d: X
n→ [0, +∞[ is an n-distance on X if it satisfies
(i) d(x
1, . . . , x
n) = 0 if and only if x
1= · · · = x
n,
(ii) d(x
1, . . . , x
n) = d(x
π(1), . . . , x
π(n)) for all x
1, . . . , x
n∈ X and all π ∈ S
n, (ii) d(x
1, . . . , x
n) ≤ P
ni=1
d(x
1, . . . , x
n)
zifor all x
1, . . . , x
n, z ∈ X ,
where we denote by d(x
1, . . . , x
n)
zithe function obtained from d(x
1, . . . , x
n) by setting its ith variable to z
For an n-distance d: X
n→ [0, +∞[, the set of the reals K of ]0, 1] for which the condition
d(x
1, . . . , x
n) ≤ K
n
X
i=1
