One of the blogs that I recently added to my RSS reader is God Plays Dice. That blog has quite a number of mathematical looks at everyday questions. One question, in particular, caught my attention: Which two states are closest together?
Your first reaction is probably Lots of states have a neighbor that’s zero miles away!
But, then, you figure that’s not a very interesting question. He must have had a more interesting question than that in mind. What was it?
Within that article, he limits this to states which do not share any boundary points (intersection of the closures of their interiors is empty).
That’s not where I thought he was going to go though. I thought he was going to measure the distance from state A to state B differently. I thought he was going to say that the distance from state A to state B is the average distance from points in state A to their nearest points in state B. In other words, the distance from state A to state B is the expected minimum distance a crow in state A would need to fly to get to state B (assuming crows are equiprobable at all spots).
This is interesting because, in particular, it is not symmetric. The distance
from state A to state B may be greater than the distance from state B to state A. Consider Louisiana and Texas, for example. In general, mathematical distance functions are required to be symmetric. I will have to explore what things break when they are not.
That’s an interesting way of reinterpreting my question. One way to do something like your distance, but symmetric, would be to say that the distance between state A and state B is the average distance between a random point in state A and a random point in state B.
In this case, determining the answer to the question “which two states are closest together?” is much more difficult. But it’s clear that one wants two small states which border each other; my guess is Connecticut-Rhode Island.
That’d be my guess, too. When I started trying to write down an integral for this notion of closeness, I accidentally started with the symmetric version you mention here.
I haven’t done much with metrics and measure in a long time and nothing beyond undergraduate-level metrics at all. I am interested in toying with them now though to see what breaks if any of the requirements are violated. I know some of the things that go wrong if the triangle inequality doesn’t hold. But, I never messed with symmetry before.
If you’re allowing random points in the starting state, why restrict to the nearest point in the ending state? Why not take the average distance from a random point in the starting state to a random point in the ending state?
This has two advantages. First, the resulting function is symmetric. Second, the integrals should be easier to set up and work out.
I’m not sure, but a first guess would be that this works out to be controlled by the distances between the centroids. Obvious problems with that would be extremely nonconvex states, but I don’t think we have any that are that bad.
Yes… I accidentally did set up the integral incorrectly at first to answer the random-point to random-point question. It does work out to be the distance between centroids. As such, it’s not terribly tough to calculate. But, I’m liking the idea of toying with non-symmetric distance.
It can’t be uphill both ways…. 🙂