Careful, though. There isn't a constant "pi" for all metric spaces. Using distance along the surface of a sphere, the ratio of a circle's diameter to circumference depends on the size of the circle.

Yeah, it works for the particular set of metrics considered but not for all metrics. I'm not sure how interesting this class of metrics is (Euclidean and Manhattan are interesting in their own right, but the others?), and I suppose that bears on how interesting this result is.