## On monotone minimal cuscos.(English)Zbl 1052.54020

A multifunction $$F$$ taking as values subsets of the real line is called $$lu^d$$-minimal if and only if for arbitrary densely defined selectors $$f_1$$, $$f_2$$ for $$F$$ the lower and resp. upper hulls coincide: $$l(f_1)= l(f_2)$$, $$u(f_1)= u(f_2)$$. If $$X$$ is a Banach space, then a multifunction $$F: A\to X^*$$, $$A\subseteq X$$ is $$lu^d$$-minimal if the multifunction: $A\ni X\mapsto F_y(x):= \{x^*(y): x^*\in F(x)\}\subseteq\mathbb{R}$ is $$lu^d$$-minimal for each $$y\in X$$, $$\| y\|= 1$$.
As an application of $$lu^d$$-minimality it is proved that an $$F: A\to X^*$$ which is a minimal $$w^*$$-cusco and has a densely defined monotone selector is monotone itself. This result in turn leads to a characterization of convexity for locally Lipschitz functions which possess a minimal subdifferential.

### MSC:

 54C60 Set-valued maps in general topology 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 52A41 Convex functions and convex programs in convex geometry
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### References:

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