  Consider a finite system of non-strict real polynomial inequalities and
suppose its solution set $S\subseteq\mathbb R^n$ is convex, has nonempty
interior and is compact. Suppose that the system satisfies the Archimedean
condition, which is slightly stronger than the compactness of $S$. Suppose that
each defining polynomial satisfies a second order strict quasiconcavity
condition where it vanishes on $S$ (which is very natural because of the
convexity of $S$) or its Hessian has a certain matrix sums of squares
certificate for negative-semidefiniteness on $S$ (fulfilled trivially by linear
polynomials). Then we show that the system possesses an exact Lasserre
relaxation.
  In their seminal work of 2009, Helton and Nie showed under the same
conditions that $S$ is the projection of a spectrahedron, i.e., it has a
semidefinite representation. The semidefinite representation used by Helton and
Nie arises from glueing together Lasserre relaxations of many small pieces
obtained in a non-constructive way. By refining and varying their approach, we
show that we can simply take a Lasserre relaxation of the original system
itself. Such a result was provided by Helton and Nie with much more machinery
only under very technical conditions and after changing the description of $S$.
