  Let X be a real nondegenerate projective subvariety such that its set of real
points is Zariski dense. We prove that every real quadratic form that is
nonnegative on X is a sum of squares of linear forms if and only if X is a
variety of minimal degree. This substantially extends Hilbert's celebrated
characterization of equality between nonnegative forms and sums of squares. We
obtain a complete list for the cases of equality and also a classification of
the lattice polytopes Q for which every nonnegative Laurent polynomial with
support contained in 2Q is a sum of squares.
