Average distance to the middle
The previous post looked at the average p distance between points in the p disk. This post looks at a related question, the average distance to the center. Unlike the previous post, we will look at dimension n greater than 2.
The paper [1] cited earlier ends with this statement:
It should be pointed out that J. S. Lew proved a more general result: in n-dimensional space, the average distance from a point in the unit p ball to the center is n/(n + 1) for all p.
There is a paper by by J. S. Lew listed in the references with the same author and title as [2], but in place of the journal and page number it says this Review, to appear." So perhaps the authors of [1] talked to the authors of [2] and knew that they intended to prove the result quoted above. But [2] came out a year later, and did not include results for dimensions higher than n = 2, unless I've overlooked something.
I imagine the theorem above, if it's true, is tedious to prove for general p. But we can show that it's true for p = 2.
For n = 2, we use polar coordinates. The distance to the origin is simply r, the volume element is r dr d, and the area of the unit disk is , and so the average distance to the origin is
For n = 3, we use spherical coordinates. The distance to the origin is again r, the volume element is now
and the volume of the unit ball is 4/3, and so the average distance to the origin is
Finally, for general n we use hyperspherical coordinates. In n dimensions, we have an r that ranges from 0 to 1 and a that ranges from 0 to 2 as before, and we have n-2 's that run from 0 to .
The volume element in hyperspherical coordinates is
We could find the volume of the n-sphere by integrating this, but we don't have to. The integral for the average distance will have an r term with exponent n and the integral for the volume will have an r term with exponent n-1. All the other terms not involving r are the same in both integrals, so they cancel out when we take the ratio.
The average distance calculation reduces to
which proves the claim at the top of the post for p = 2.
Related posts[1] C. K. Wong and Kai-Ching Chu. Distances in lp Disks. SIAM Review, Vol. 19, No. 2 (Apr., 1977), pp. 320-324.
[2] John S. Lew, James C. Frauenthal, and Nathan Keyfitz. On the average distances in a circular disc. SIAM Review, Vol. 20, No. 3 (Jul., 1978), pp. 584-592.