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Every convex free basic semi-algebraic set has an LMI representation. (English) Zbl 1260.14011

It is known that the matricial solution set of a linear matrix inequality (LMI) is a convex free basic open semi-algebraic set. The main theorem of this paper is a converse, namely that each such set arises from some LMI. The result has implications for semi-definite programming and systems engineering as well as for free semi-algebraic geometry.

MSC:

14C21 Pencils, nets, webs in algebraic geometry
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A40 Inequalities and extremum problems involving convexity in convex geometry
14K05 Algebraic theory of abelian varieties
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