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 Sion’s minimax result. Although this proof technique is initiated by Joó and based on the inter-section 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 simplified the original proof of Joó and give a more elementary proof of the celebrated Sion’s minimax theorem.