  The compact set of homogeneous quadratic polynomials in $n$ real variables
with modulus bounded by 1 on the unit sphere $S^{n-1}$ is trivially
semi-definite representable. The compact set of homogeneous ternary quartics
with modulus bounded by 1 on the unit sphere $S^2$ is also semi-definite
representable. This suggests that the compact set of homogeneous ternary cubics
with modulus bounded by 1 on $S^2$ is semi-definite representable. We deduce an
explicit semi-definite representation of this norm ball. More generally, we
provide a semi-definite description of the cone of inhomogeneous ternary cubics
which are nonnegative on $S^2$.
