(Replying to PARENT post)

There is no concept of the "background metric" here. Both the radius and the circumference are measured in the defined metric itself.

Any metric that "pulls on the origin" compared to Euclidean distance will have to do the mapping in a continuous way. This will basically result in both the radius and circumference being expanded in that metric.

Matter of fact, I linked an article that proves that for _all_ metrics, the value of ฯ€ is always between 3 and 4 (inclusive). Unfortunately the article might have gotten the hug of death so here is an alternative link: https://www.researchgate.net/publication/353330827_Extremal_...

๐Ÿ‘คazeemba๐Ÿ•‘2y๐Ÿ”ผ0๐Ÿ—จ๏ธ0

(Replying to PARENT post)

How is circumference defined?

And I can think of a counterexample on a sphere, just using Euclidean distance on the surface. Consider a circle with centre at North Pole and radius being the distance from the North Pole to a point on the equator. For this circle it is easy to find out that pi=2

๐Ÿ‘คcharlieyu1๐Ÿ•‘2y๐Ÿ”ผ0๐Ÿ—จ๏ธ0