Question

What is the average length of a chord on the unit sphere? (e.g. straight-line distance between two randomly-chosen points) [3 weeks unsolved]

Answer 1

Lets take a look at he simpler problem involving a unit circle:

Here I have chosen some point P on the circumference and then drawn several chords from P to another point on the circumference.
I would observe that there are an equal number on points from P to the circumference above the diameter and below the diameter. Therefore, on average, I would say the chord length would just be the diameter of the circle which is 2.

My intuition tells me the same should hold for a unit sphere.