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It was just an example.what I am expecting is a structure with a metric $d_n:X^n\rightarrow \mathbb R$ with properties similar to the metric $d:X\times X\rightarrow \mathbb R$.And presently I am working on some possibilities.
@Paul..I have developed a way to generalize the metric to $n$ variables and used it to generalize the concept of bitopological spaces but I am not sure whether such generalizations deserve publication.I am also not sure about any such journal where I could publish my work.
@james what if we still want our generalized metric to have the notion of distance. for example we can take $d(x,y,z)$ as the sum of the sides of the triangle with vertices $x,y,z$.
@ Asaf $X$ is a non-empty set, and $d:X^n\rightarrow \mathbb R$ be a function we want to make it a metric(analogue) and here we are discussing whether such properties could be defined for $d$.
@ chandersekher... Also Z. Mustafa B. Sims, A new approach to generalized metric spaces, J. Of Nonlinear and Convex Analysis, 7 (2006), No.2, 289-297.do you still think that it is not a research-level question?
@chandrasekhar I am working on the generalization of metric spaces with n-variables. for three variables we have research work [see S. Gähler, 2-metrische räume und ihre topologische struktur, Math. Nachr., 26 (1963), 115-148., Gähler, Zur geometric 2-metrische räume, Rev. Roum. Math. Pures et Appl., 11 (1966), 664-669.,Z. Mustafa and B. Sims, Some remarks concerning D-metric spaces, Proceedings of the International Conferences on Fixed Point Theory and Applications, Valencia (Spain), July (2003), 189-198.,....]
