Let us define $d:X^n\rightarrow\mathbb R$. How can we define metric properties such as symmetry, triangle inequality equivalent property etc for such a function?
closed as not a real question by Asaf Karagila, Andreas Blass, Steven Landsburg, Andrés E. Caicedo, Ryan Budney Jun 21 '12 at 18:07It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question. 


One should let the properties follow from an interpretation. We didn't write down the axioms for a metric space because they're intrinsically interesting: we wrote them down because they describe the notion of distance. One thing that certainly has been tried is to let $d(x,y,z)$ describe the area of a triangle between the three points $x,y,z$. Then the appropriate axioms are that $d$ be completely symmetric (swapping any two of $x$, $y$ and $z$ gives the same answer) and that $$d(x,y,z) \leq d(a,y,z) + d(x,a,z) + d(x,y,a).$$ In general, one could let $d(x_0,\ldots,x_n)$ be thought of as the $n$volume of an $n$simplex. Then the appropriate axioms are complete symmetry again, and a very similar triangle inequality (where the righthandside is the sum of what you get by substituting $a$ for each $x_i$). 


Here are some comments about why one might be interested in generalizing metrics to multiple inputs. These comments, however, seem somewhat orthogonal to what the OP is looking for.



There are quite a few people who have tried to generalize metrics to more than two variables. I once tried to track down all the references on this subject for a paper. Here are some: There is an extensive literature on 2metrics, in which $d$ takes 3 arguments. This appears to have been introduced by Gahler. Here is a recent example with some references. What James Cranch mentions in his answer is (I think) originally due to Menger (K. Menger, Untersuchungen uber allgemeine Metrik, Math. Ann. 100.). Menger takes $d$ to be the volume of an $n$simplex in Euclidean space. Then he tries to abstract away from that. (I can't read German so take this summary with a grain of salt.) Three recent papers that seek such generalizations are by Deza and Rosenberg (Small cones of $m$hemimetrics), by Chepoi and Fichet (A note on threeway dissimilarities and their relationship with twoway dissimilarities), and by Warren ($n$way metrics). My impression from all of these is that there is no one natural way to extend metrics to take more than 2 arguments. 

