Semidefinite representation for convex hulls of real algebraic curves

2018
Journal article
Published
Published in
SIAM Journal on Applied Algebra and Geometry ; 2 (2018), 1. - pp. 1-25. - eISSN 2470-6566
Abstract
We show that the closed convex hull of any one-dimensional semialgebraic subset of $\mathbb{R}^n$ is a spectrahedral shadow, meaning that it can be written as a linear image of the solution set of some linear matrix inequality. This is proved by an application of the moment relaxation method. Given a nonsingular affine real algebraic curve $C$ and a compact semialgebraic subset $K$ of its $\mathbb{R}$-points, the preordering $\mathscr{P}(K)$ of all regular functions on $C$ that are nonnegative on $K$ is known to be finitely generated. Our main result, from which all others are derived, says that $\mathscr{P}(K)$ is stable, meaning that uniform degree bounds exist for weighted sum of squares representations of elements of $\mathscr{P}(K)$. We also extend this last result to the case where $K$ is only virtually compact. The main technical tool for the proof of stability is the archimedean local-global principle. As a consequence of our results we show that every convex semialgebraic subset of $\mathbb{R}^2$ is a spectrahedral shadow.
510 Mathematics
Keywords
spectrahedral shadows, convex algebraic geometry, real algebraic curves, convex hull, linear matrix inequalities, moment relaxation, semidefinite programming, Helton-Nie conjecture
Cite This
ISO 690SCHEIDERER, Claus, 2018. Semidefinite representation for convex hulls of real algebraic curves. In: SIAM Journal on Applied Algebra and Geometry. 2(1), pp. 1-25. eISSN 2470-6566. Available under: doi: 10.1137/17M1115113
BibTex
@article{Scheiderer2018Semid-23348.2,
year={2018},
doi={10.1137/17M1115113},
title={Semidefinite representation for convex hulls of real algebraic curves},
number={1},
volume={2},
journal={SIAM Journal on Applied Algebra and Geometry},
pages={1--25},
author={Scheiderer, Claus}
}

RDF
<rdf:RDF
xmlns:dcterms="http://purl.org/dc/terms/"
xmlns:dc="http://purl.org/dc/elements/1.1/"
xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
xmlns:bibo="http://purl.org/ontology/bibo/"
xmlns:dspace="http://digital-repositories.org/ontologies/dspace/0.1.0#"
xmlns:foaf="http://xmlns.com/foaf/0.1/"
xmlns:void="http://rdfs.org/ns/void#"
xmlns:xsd="http://www.w3.org/2001/XMLSchema#" >
<dcterms:isPartOf rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
<foaf:homepage rdf:resource="http://localhost:8080/"/>
<dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2018-09-10T11:54:49Z</dcterms:available>
<dcterms:rights rdf:resource="https://rightsstatements.org/page/InC/1.0/"/>
<dc:contributor>Scheiderer, Claus</dc:contributor>
<dc:creator>Scheiderer, Claus</dc:creator>
<dcterms:hasPart rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/23348.2/1/Scheiderer_2-1n6fa7dg51tl00.pdf"/>
<dc:language>eng</dc:language>
<dcterms:issued>2018</dcterms:issued>
<dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2018-09-10T11:54:49Z</dc:date>
<void:sparqlEndpoint rdf:resource="http://localhost/fuseki/dspace/sparql"/>
<dspace:isPartOfCollection rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
<dcterms:title>Semidefinite representation for convex hulls of real algebraic curves</dcterms:title>
<bibo:uri rdf:resource="https://kops.uni-konstanz.de/handle/123456789/23348.2"/>
<dspace:hasBitstream rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/23348.2/1/Scheiderer_2-1n6fa7dg51tl00.pdf"/>
<dcterms:abstract xml:lang="eng">We show that the closed convex hull of any one-dimensional semialgebraic subset of $\mathbb{R}^n$ is a spectrahedral shadow, meaning that it can be written as a linear image of the solution set of some linear matrix inequality. This is proved by an application of the moment relaxation method. Given a nonsingular affine real algebraic curve $C$ and a compact semialgebraic subset $K$ of its $\mathbb{R}$-points, the preordering $\mathscr{P}(K)$ of all regular functions on $C$ that are nonnegative on $K$ is known to be finitely generated. Our main result, from which all others are derived, says that $\mathscr{P}(K)$ is stable, meaning that uniform degree bounds exist for weighted sum of squares representations of elements of $\mathscr{P}(K)$. We also extend this last result to the case where $K$ is only virtually compact. The main technical tool for the proof of stability is the archimedean local-global principle. As a consequence of our results we show that every convex semialgebraic subset of $\mathbb{R}^2$ is a spectrahedral shadow.</dcterms:abstract>
<dc:rights>terms-of-use</dc:rights>
</rdf:Description>
</rdf:RDF>

Yes
Unknown

Version History

Now showing 1 - 2 of 2
VersionDateSummary
2*
2018-08-27 09:32:04
2013-06-04 10:14:28
* Selected version