  We prove that convex ternary quartic forms are sum-of-squares-convex
(sos-convex). This result is in a meaningful sense the ``convex analogue'' a
celebrated theorem of Hilbert from 1888, where he proves that nonnegative
ternary quartic forms are sums of squares. We show by an appropriate
construction that exploiting the structure of the Hessian matrix is crucial in
any possible proof of our result.
