# How to solve this mathematical programming problem?

Suppose each point in the interval $$[0,1]$$ has some of a set of $$N$$ properties. The total length of the intervals with property $$i$$ is $$x_i$$ ($$0\le x_i \le 1$$).

Questions:

1. What is the maximum and minimum of the length of intervals of points with exactly $$m$$ properties?
2. Is there some none-zero-length interval with $$m$$ properties? This second question may be seen as part of the first one.

Not sure about what is the proper name of such kind of problems.

An example:

In a school, 80% of the students play basketball, 60% play football, 60% play tennis, 60% play baseball.

Question: Is it true that, there must be some students play all four of the sports? What is the maximum and minimum fraction of students play three of the four sports?

Introduce $$2^4=16$$ nonnegative decision variables, one for each subset of the four sports. They must sum to $$1$$, and the given percentages imply four additional linear constraints.