Dimovski D.  

(n,ρ,m) metric spaces

    For positive integers n,m with n>m and a subset ρ of the n^th symmetric power M⁽ⁿ⁾ of a set M, such that
Δn={(x,..., x)_n | x∈M}⊆ρ, a map d from M⁽ⁿ⁾ into the set [0,∞) of nonnegative real numbers is said to be an (n,ρ,m) metric on M, if for each x∈M⁽ⁿ⁾ :
(1) d(x) = 0 if and only if x∈ρ; and
(2) for each u∈M^(m) , d(x) ≤ Σ d(yu), where the sum is over all y∈M^(n–m) such that x = yv for some v∈M^(m).
     With this notion, a (2,Δ₂,1) metric is the usual notion of a metric, a (2,ρ,1) metric is the notion of a pseudometric, and the notion of (n,ρ,1) metric is the notion of (n+1) metric defined by K. Menger in the paper Untersuchungen über allgemeine Metrik, Math. Ann. 100, (1928), pp. 75-163.
     We investigate the properties of (n,ρ,m) metric spaces M, i.e. the sets equipped with an (n,ρ,m) metric d, with the aim to use them for recognizing images.