There's a stackexchange thread that touches on the first two questions. It's got the integral form of the circumference calculation, but I doubt there's a close

There's a stackexchange thread that touches on the first two questions. It's got the integral form of the circumference calculation, but I doubt there's a closed-form solution in general: https://math.stackexchange.com/questions/2044223/measuring-pi-with-alternate-distance-metrics-p-norm https://math.stackexchange.com/questions/2044223/measuring-p...