Sinn, Rainer Algebraic boundaries of \(\mathrm{SO}(2)\)-orbitopes. (English) Zbl 1285.14062 Discrete Comput. Geom. 50, No. 1, 219-235 (2013). The algebraic boundary of a semi-algebraic set is an object of great interest in convex algebraic geometry and semi-definite optimization. In this paper, it is used to study whether or not a semi-algebraic set is basic closed. A convex semi-algebraic set is basic closed if it can be described by finitely many polynomial inequalities. For instance, a spectrahedron is a basic closed semi-algebraic set. An interesting class of compact convex semi-algebraic sets are the orbitopes. An orbitope is the convex hull of an orbit under a linear action of a compact real algebraic group on a real vector space, see [R. Sanyal, F. Sottile and B. Sturmfels, Mathematika 57, No. 2, 275–314 (2011; Zbl 1315.52001)]. The focus of this paper is on the special case of orbitopes of the group \(\mathrm{SO}(2)\) of real orthogonal \(2\times 2\) matrices with determinant \(1\). The main result is the following: Let \(X\subset \mathbb A^{2r}\) be an irreducible curve and assume that the real points \(X(\mathbb R)\) of \(X\) are dense in \(X\) in the Zariski topology. Let \(C\) be the convex hull of \(X(\mathbb R)\subset \mathbb R^{2r}\) and suppose that the interior of \(C\) is non-empty. Then the \((r-1)\)th secant variety to \(X\) is an irreducible component of the algebraic boundary of \(C\) if and only if the set of all \(r\)-tuples of real points of \(X\) that span a face of \(C\) has dimension \(r\). As applications of this result, it is proved that the Barvinok-Novik orbitopes are not basic closed and it is given a characterization for basic closed \(4\)-dimensional \(\mathrm{SO}(2)\)-orbitopes. Reviewer: Emanuele Ventura (Helsinki) Cited in 7 Documents MSC: 14P05 Real algebraic sets 52A99 General convexity 90C22 Semidefinite programming 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 14Q05 Computational aspects of algebraic curves Keywords:convex hull; secant variety; basic closed semi-algebraic set Citations:Zbl 1315.52001 Software:Macaulay2 PDFBibTeX XMLCite \textit{R. Sinn}, Discrete Comput. Geom. 50, No. 1, 219--235 (2013; Zbl 1285.14062) Full Text: DOI arXiv References: [1] Barvinok, A., Novik, I.: A centrally symmetric version of the cyclic polytope. Discrete Comput. Geom. 39(1-3), 76-99 (2008) · Zbl 1184.52010 [2] Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 10, 2nd edn. Springer-Verlag, Berlin (2006) · Zbl 1102.14041 [3] Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36. 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