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Extensions of cyclically monotone mappings. (English) Zbl 0647.90071
The authors study cyclically monotone mappings and the subclass of them called strongly cyclically monotone (s.c.m.) mappings (the mapping $$f: S\subset R$$ $$n\to R^ n$$ is said to be s.c.m. on S if $$\sum^{k}_{i=1}(x_ i-x_{i-1})f(x_ i)\geq 0$$ for all $$k\geq 2$$ and $$x_ 1,...,x_ k=x_ 0\in S$$, and $$\sum^{k}_{i=1}(x_ i-x_{i- 1})f(x_ i)=0$$ implies that $$f(x_ 1)=...=f(x_ k)=0$$ for all $$k\geq 2$$ and every $$x_ 1,...,x_ k=x_ 0\in S)$$. They prove the following main result: Let $$f: S\subset R$$ $$n\to R^ n$$ be s.c.m. Then f can be extended to some s.c.m. mapping $$\bar f$$ defined on conv S (the convex hull of S).

##### MSC:
 90C25 Convex programming 49J35 Existence of solutions for minimax problems 90C31 Sensitivity, stability, parametric optimization