I have come to know from Wikipedia article about what are called generalized metrics, and that they differ from the regular metric definition in terms of the properties/requirements they have to satisfy. Hence I'd like to know if there is any metric used in literature, and its pros and cons for used/not being used with any set with following properties.
Properties(assume a a set $M$.
$d(a,b) \in (1,\infty) \forall a,b \in M$.
$d(a,b) = d(b,a)$.
$d(a,b) = 1$, then $a=b$
$d(a,b) d(a,c) > d(b,c)$.
The fourth condition should be $d(a,b) d(a,c) \ge d(b,c)$; otherwise it contradicts 3 when $a=b$, and generally rules out some interesting things.
The standard example of a function $d$ with the above properties is the multiplicative Banach-Mazur distance between $n$-dimensional normed spaces. Even though it could be turned into ordinary metric by taking logarithm (as Anthony Carapetis commented), in practice the logarithm only clutters the computations.
One can consider the Banach-Mazur distance between infinite-dimensional Banach spaces, in which case it becomes $\infty$ for non-isomorphic spaces. Put differently, the metric is finite within each isomorphism class. It is an open question whether every isomorphism class is unbounded in this metric. For separable spaces this was proved not long ago, in
Johnson, W. B., Odell, E., The diameter of the isomorphism class of a Banach space, Ann. Math. 162 (2005), 423-437.
