What Do You Mean By Close? May 18th, 2009
Patrick Stein

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.

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