In a previous post on this topic, I said that hyperspheres get a bad rap. They’re doing their best to be perfectly round, and someone comes along and accuses them of being inadequate, or weird. It turns out that hyperspheres aren’t really weird at all. It’s measure that’s weird. And where measure is concerned, there are objects out there that truly display that weirdness.
To recap, I was talking about how the volume of a unit hypersphere measured the normal way (with its radius = 1) approaches zero with increasing dimension. I also mentioned that even though a “unit” hypercube that circumscribes the unit sphere (i.e., a hypercube with inradius = 1) has volume that increases exponentially with the dimension (2d), a hypercube with circumradius = 1 decreases even faster than the volume of the hypersphere. Why is one configuration different than the other?
The answer is that they’re not different. A cube is a cube, no matter how you orient it. If its side is of length s, then its volume is sd. What’s different here is our notion of unit measure. We commonly define a unit of volume as the volume of a hypercube with sides of unit length. In that light, it’s not terribly surprising what we know about the volume of hypercubes. So why can’t we just define the unit hypersphere to have unit volume?
This seems objectionable until you realize that we do this all the time in the real world. What’s a gallon? It has nothing to do with an inch or foot. So why do we worry ourselves over defining volume in terms of one-dimensional units? The metric system doesn’t even adhere to this standard. A liter is a cubic decimeter. Why? It just worked out that way. Since these units are all just arbitrary, we could just declare that unit volume is the volume of a unit hypersphere. Or not. So a hypersphere’s volume really isn’t that weird. What seems weird is the discrepancy between the geometries of the hypercube and hypersphere.
Are there objects that do act strangely in higher dimensions? Definitely. Consider a multivariate normal distribution (a Gaussian distribution in multiple dimensions). For the sake of simplicity, I’ll consider one with zero mean and variance σ2:
Multivariate Gaussians are all nice and round. What can we say about what the distance from the mean (0) looks like? Well, this is just the variance:
How much does it deviate from this value? We can apply a Chernoff bound (don’t ask me how; deriving Chernoff bounds is not my strong suit):
Let’s take another look at this bound, though. It’s saying that the probability of the squared distance from the mean deviating from nσ2 by more than a small percentage decreases exponentially with n. So the points that follow the distribution mostly sit in a thin shell around the mean. But the density function still says that the density is highest at the mean. Now that’s weird.
Why does this happen, though? It’s difficult to get a handle on, but the word “density” is what you have to pay attention to. That shell has an incredibly high volume at higher dimensions (it grows with drd-1). High enough that the density is still lower at the shell than at the mean. Why it’s highest in the shell is even more difficult to figure out. I don’t have a good answer, but I suspect that it has something to do with the fact that the distribution must add up to one, and it has to “fill” all the nooks, and it can’t do that at the mean. It has to do this out in this thin shell.
[Update: I guessed last night that if I multiply the p.d.f. by the boundary volume (i.e. "surface area") of a hypersphere of radius ||x||, then I should see spikes out at σ√n. I was correct. Below, Micheal Lugo confirmed that intuition slightly more rigorously in the comments. He's a probabilist, so I think I'm safe. :-) ]
There are certainly weird things that happen in higher dimensions. In my opinion, all these things have more to do with measure than geometry.