1 Max-min convexity

The max-min semifield is the set \overline{R} = R ∪ \{−∞, +∞\} endowed with the operations ⊕ = max, ⊗ = min. We study the semimodule \overline{R}^{n} = \overline{R} × ... × \overline{R} (n times), with the operations ⊕ and ⊗ defined componentwise. A subset G of \overline{R}^{n} (respectively, a function f : \overline{R}^{n} → \overline{R}) is said to be max-min convex if the relations x, y ∈ G (respectively, x, y ∈ \overline{R}^{n}) and α, β ∈ \overline{R}, α ⊕ β = +∞, where +∞ is the neutral element for ⊗ = min, imply (α ⊗ x) ⊕ (β ⊗ y) ∈ G (respectively, f((α ⊗ x) ⊕ (β ⊗ y)) ≤ (α ⊗ f(x)) ⊕ (β ⊗ f(y)). We give some new results on max-min convexity of sets and functions in \overline{R}^{n} (e.g. on segments, semispaces, separation, multi-order convexity, ...) that correspond to some results for max-plus convexity, replacing ⊗ = + of the max-plus case by the semi-group operation ⊗ = min of the max-min case. References K. Zimmermann, Convexity in semimodules. Ekonom.-Mat. Obzor 17 (1981), 199-213. V. Nitica and I. Singer, Contributions to max-min convex geometry. I: Segments. Lin. Alg. Appl. 428 (2008), 1439-1459. II: Semispaces and convex sets. Ibidem 2085-2115.