The level set method of Joó and its use in minimax theory

In this paper we discuss the level set method of Joó and how to use it to give an elementary proof of the well-known minimax theorem of Sion. Although this proof technique was initiated by Joó and based on the intersection of upper level sets and a clever use of the topological notion of connectedness, it is not very well known and accessible for researchers in optimization. At the same time we simplify the original proof of Joó and give a more elementary proof of the celebrated minimax theorem of Sion.