The previous post looked at the average ℓ_{p} distance between points in the ℓ_{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 ℓ_{p} ball to the center is *n*/(*n* + 1) for all *p*.

There is a paper 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*θ, and the area of the unit disk is π, 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 4π/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 θ that ranges from 0 to 2π as before, and we have *n*-2 φ’s that run from 0 to π.

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 l_{p} 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.

I used to read all of your posts via email. Now that you have switched, the emails only show a snippet of the post (one or two sentences), making me click a link to read the full post. This takes me out of my email client and discourages me from reading. Since the switch, I have found that I read 10% of the posts. Can you make it to where the email includes the full blog post?

In fact, you can apply a similar argument for any p, or any norm for that matter. All you need is that the volume of a ball of radius r is proportional to r^n, which follows from the linearity property of a norm (that is, a ball of radius r is just a ball of radius 1 scaled by a factor of r, regardless of what that ball looks like an a particular norm).

@Austin: I’m sorry but I don’t believe I have that option.

@Nathan: That’s slick.