Which Interval Is the Closest to a Given Set?

In some practical situations, we know a set of possible values of a physical quantity – a set which is not an interval. Since computing with sets is often complicated, it is desirable to approximate this set by an easier-to-process set: namely, with an interval. In this paper, we describe intervals which are the closest approximations to a given set. 1 Formulation of the Problem Why do we need 1-D sets beyond intervals. For each physical quantity, we would like to know which real numbers are possible values of this quantity. For some quantities, the possible values come from fundamental physics. For example, the possible values of velocity form an interval [0, c], where c is the speed of light. For other quantities – e.g., for size of some insect species – we have to rely on experts. An expert usually provides us with a possible interval range. Different experts can provide different range. Some of these ranges may be non-intersecting: for example, one expert provides a range of adult cockroach sizes in St. Petersburg, Russia, where they are reasonably small, while another expert provides a range of adult cockroach sizes in El Paso, Texas, where the cockroaches are much larger. If we trust all the experts, then we consider all the values supplied by all the experts as possible values of the corresponding quantity. Thus, after asking all the experts, we end up with the union of their answer sets as the desired set of possible values of the quantity – and this union of intervals is not necessarily an interval itself: • In some cases, if two values a < b are possible, then all intermediate values from the interval [a, b] are also possible. • However, this is not always the case: when a species consists of two populations of different sizes, the set of possible sizes is the union of two intervals; see, e.g., [5] and references therein.