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\section { Max-min convexity}  

\medskip  \noindent The max-min semifield is the set $\overline{R}=R\cup \{-\infty ,+\infty \}$ endowed with the operations $\oplus =\max ,\otimes =\min $. We study the semimodule $\overline{R}^{n}=\overline{R}\times ...\times \overline{R}$ ($n$ times), with the operations $\oplus $ and $\otimes $ defined componentwise. A subset $G$ of $\overline{R}^{n}$ (respectively, a function $f:\overline{R} ^{n}\rightarrow \overline{R}$) is said to be max-min convex if the relations $x,y\in G$ (respectively, $x,y\in \overline{R}^{n}$) and $\alpha ,\beta \in \overline{R}$, $\alpha \oplus \beta =+\infty $, where $+\infty $ is the neutral element for $\otimes =\min $, imply $(\alpha \otimes x)\oplus (\beta \otimes y)\in G$ (respectively, $f((\alpha \otimes x)\oplus (\beta \otimes y))\leq (\alpha \otimes f(x))\oplus (\beta \otimes f(y)$).  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 $\otimes =+$ of the max-plus case by the semi-group operation $ \otimes =\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.
%Max-min convex set; Max-min convex function","08A72","52A01","","03:16:01","Thu Mar 13 2008","92.80.46.206"
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