Partially non-convex minimax theorem and applications to unique remoteness of sets and functions

We establish the following variant of the minimax inequality maxy∈Binfx∈Af(y,x)≥infx∈Asupy∈B0f(y,x), where, unlike the classical minimax theorem, the functions f(y,⋅) may not be convex for all y∈B but only for some of them (those in a possibly smaller set B0), while the function f is allowed to take infinite values. An application to the unique remoteness of sets and functions is given.