How to calculate the distance between two (possibly unbounded) ranges of positive real numbers? For example, if three guys specify their prices they would pay for a product:
A would pay between 2€ and 10€ B would pay less than 6€ C would pay more than 8€
How to say if A's offer is more similar to B's or C's? If all ranges are bounded, I could use the Hausdorff distance. But what if one or both of the sets are unbounded? Note that they can be unbounded only from one side (all numbers are positive).
EDIT: To explain why I need this... I'm working on a clustering algorithm to group similar objects that are represented by ranges of real numbers. Let's say that I have a group of people who are buying sugar. Each of them defines minimum and maximum amount he would be willing to buy. What I would like to do is to put in one cluster those people that want to buy similar amount. For example, one of the result clusters would be for people who need between 3 and 4 kg of sugar, while the other one might be between 6 and 8 kg. For this, I have to create a distance function which would tell how similar the requirements of the buyers are, i.e., what is the distance (or difference) between two ranges.