  Motivated by scheme theory, we introduce strong nonnegativity on real
varieties, which has the property that a sum of squares is strongly
nonnegative. We show that this algebraic property is equivalent to
nonnegativity for nonsingular real varieties. Moreover, for singular varieties,
we reprove and generalize obstructions of Gouveia and Netzer to the convergence
of the theta body hierarchy of convex bodies approximating the convex hull of a
real variety.
