# Convex Algebraic Geometry of Curvature Operators

## TL;DR

This paper explores the convex algebraic geometry of curvature operators, identifying when certain curvature sets are spectrahedra, providing counterexamples to a conjecture, and developing algorithms for membership testing.

## Contribution

It characterizes the convex semialgebraic sets of curvature operators as spectrahedra or spectrahedral shadows and offers algorithms for membership testing, including counterexamples to the Helton--Nie Conjecture.

## Key findings

- For dimensions n≥5, the set is not a spectrahedron, providing counterexamples to the Helton--Nie Conjecture.
- Efficient algorithms for n=4 to test membership in the curvature set.
- Semidefinite programming algorithms for n≥5 using hierarchies of spectrahedral shadows.

## 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.