Type of Publication:  Journal article 
Publication status:  Published 
URI (citable link):  http://nbnresolving.de/urn:nbn:de:bsz:3522dgslzdhlcqfr7 
Author:  Scheiderer, Claus 
Year of publication:  2018 
Published in:  SIAM Journal on Applied Algebra and Geometry ; 2 (2018), 1.  pp. 125.  eISSN 24706566 
ArXivID:  arXiv:1208.3865 
DOI (citable link):  https://dx.doi.org/10.1137/17M1115113 
Summary: 
We show that the closed convex hull of any onedimensional 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 localglobal principle. As a consequence of our results we show that every convex semialgebraic subset of $\mathbb{R}^2$ is a spectrahedral shadow.

Subject (DDC):  510 Mathematics 
Keywords:  spectrahedral shadows, convex algebraic geometry, real algebraic curves, convex hull, linear matrix inequalities, moment relaxation, semidefinite programming, HeltonNie conjecture 
Link to License:  Terms of use 
Bibliography of Konstanz:  Yes 
Refereed:  Unknown 
SCHEIDERER, Claus, 2018. Semidefinite representation for convex hulls of real algebraic curves. In: SIAM Journal on Applied Algebra and Geometry. 2(1), pp. 125. eISSN 24706566. Available under: doi: 10.1137/17M1115113
@article{Scheiderer2018Semid23348.2, title={Semidefinite representation for convex hulls of real algebraic curves}, year={2018}, doi={10.1137/17M1115113}, number={1}, volume={2}, journal={SIAM Journal on Applied Algebra and Geometry}, pages={125}, author={Scheiderer, Claus} }
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