But remember that "we" chose the generalization which this all depends on.

Well, it's pi parameterised by the distance metric, Pi(d)

You can parameterise it by other concerns if you wish, and other things follow. But as a matter of fact, this is how pi depends on the distance metric.

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amelius a day ago | parent [-]

Yes, but we're looking at a specific set of distance metrics.

It'd be interested in the set where Pi(d) is constant and equal to Pi.

(disclaimer: IANAM and I haven't given it much thought)

	▲	frotaur a day ago \| parent \| next [-]
		This is certainly an interesting question; however, it's probably very hard. This is because the 'set of all possible metrics of the 2d plane' is extremely large, and I am not sure if we have a good characterization for it. There are a bunch of very strange metrics, e.g. a metric for which d(x,x) = 0, d(x,y)=1, that is, all points are at a distance of 1 to each other (this satisfies all axioms).
	▲	srean a day ago \| parent \| prev [-]
		You might like the discussion here https://math.stackexchange.com/questions/254620/pi-in-arbitr...