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arXiv · 1908.03713

Convex Algebraic Geometry of Curvature Operators

Renato G. Bettiol·Mario Kummer·Ricardo A. E. Mendes
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Abstract

We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry. More precisely, we determine in which dimensions $n$ this convex semialgebraic set is a spectrahedron or a spectrahedral shadow; in particular, for $n\geq5$, these give new counter-examples to the Helton--Nie Conjecture. Moreover, efficient algorithms are provided if $n=4$ to test membership in such a set. For $n\geq5$, algorithms using semidefinite programming are obtained from hierarchies of inner approximations by spectrahedral shadows and outer relaxations by spectrahedra.

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Renato G. Bettiol, Mario Kummer, Ricardo A. E. Mendes. 2019-08-10. Convex Algebraic Geometry of Curvature Operators. https://doi.org/10.1137/20m1350777

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