Closing the gap around the essential minimum of height functions with linear programming

For many common height functions, it is notoriously hard to compute the essential minimum. Nevertheless there are two classical methods, one giving lower bounds and the other giving upper bounds. In this paper, we show that the two methods are actually dual to each other in the sense of linear programming. The main theorem is that they satisfy strong duality, which closes the gap around the essential minimum from both ends. As applications we prove that this essential minimum can be realized by a generic sequence of algebraic integers, and that if the associated Green function is computable then this essential minimum is a computable real number.