An interval version of separation by semispaces in max–min convexity

In this paper, we study separation of a closed box from a max–min convex set by max–min semispaces. This can be regarded as an interval extension of the known separation results. We give a constructive proof of the separation in the case when the box satisfies a certain condition, and we show that the separation is never possible when the condition is not satisfied. We also study the separation of two max–min convex sets by a box and by a box and a semispace.