  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.
