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,Δ_{2},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.

Abstracts file: | n,ro,m1.pdf |

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