## A generalization of Dubovitskij-Milyutin theorem.(English)Zbl 0999.49005

From the text: “One of the fundamental theorems of convex analysis is the Dubovitskij-Milyutin theorem. Let $$X$$ be a Hausdorff locally convex space, let $$K_1,\dots, K_n$$ be convex cones in $$X$$ (with the vertex at $$0$$), all but one open, and let the intersection of all $$n$$ cones be empty. Then there exist elements $$x^*_1\in K^*_1,\dots, x^*_n\in K^*_n$$, not all zero, such that $$x^*_1+\cdots+ x^*_n= 0$$. (Here $$K^*$$ denotes the dual (polar) cone to $$K$$.)
This theorem is evidently nonsymmetric: one of the cones stands by itself. Besides it is supposed in the theorem that almost all cones are “solid”.
We give a generalization of the theorem, which is symmetric and works even in cases where none of the cones is solid”.

### MSC:

 49J27 Existence theories for problems in abstract spaces 49J53 Set-valued and variational analysis 46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
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