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# Two Results on the Size of Spectrahedral Descriptions

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KUMMER, Mario, 2016. Two Results on the Size of Spectrahedral Descriptions. In: SIAM Journal on Optimization. 26(1), pp. 589-601. ISSN 1052-6234. eISSN 1095-7189

@article{Kummer2016-02-25Resul-34059, title={Two Results on the Size of Spectrahedral Descriptions}, year={2016}, doi={10.1137/15M1030789}, number={1}, volume={26}, issn={1052-6234}, journal={SIAM Journal on Optimization}, pages={589--601}, author={Kummer, Mario} }

<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:bibo="http://purl.org/ontology/bibo/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:dcterms="http://purl.org/dc/terms/" xmlns:xsd="http://www.w3.org/2001/XMLSchema#" > <rdf:Description rdf:about="https://kops.uni-konstanz.de/rdf/resource/123456789/34059"> <dcterms:issued>2016-02-25</dcterms:issued> <dcterms:title>Two Results on the Size of Spectrahedral Descriptions</dcterms:title> <dc:language>eng</dc:language> <dcterms:abstract xml:lang="eng">A spectrahedron is a set defined by a linear matrix inequality. Given a spectrahedron, we are interested in the question of the smallest possible size $r$ of the matrices in the description by linear matrix inequalities. We show that for the $n$-dimensional unit ball $r$ is at least $\frac{n}{2}$. If $n=2^k+1$, then we actually have $r=n$. The same holds true for any compact convex set in $\mathbb{R}^n$ defined by a quadratic polynomial. Furthermore, we show that for a convex region in $\mathbb{R}^3$ whose algebraic boundary is smooth and defined by a cubic polynomial, we have that $r$ is at least five. More precisely, we show that if $A,B,C \in {Sym}_r(\mathbb{R})$ are real symmetric matrices such that $f(x,y,z)=\det(I_r+A x+B y+C z)$ is a cubic polynomial, then the surface in complex projective three-space with affine equation $f(x,y,z)=0$ is singular.</dcterms:abstract> <bibo:uri rdf:resource="https://kops.uni-konstanz.de/handle/123456789/34059"/> <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2016-05-23T09:39:43Z</dcterms:available> <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2016-05-23T09:39:43Z</dc:date> <dc:creator>Kummer, Mario</dc:creator> <dc:contributor>Kummer, Mario</dc:contributor> </rdf:Description> </rdf:RDF>