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Consider a finite system of non-strict real polynomial inequalities andsuppose its solution set $S\subseteq\mathbb R^n$ is convex, has nonemptyinterior and is compact. Suppose that the system satisfies the Archimedeancondition, which is slightly stronger than the compactness of $S$. Suppose thateach defining polynomial satisfies a second order strict quasiconcavitycondition where it vanishes on $S$ (which is very natural because of theconvexity of $S$) or its Hessian has a certain matrix sums of squarescertificate for negative-semidefiniteness on $S$ (fulfilled trivially by linearpolynomials). Then we show that the system possesses an exact Lasserrerelaxation.  In their seminal work of 2009, Helton and Nie showed under the sameconditions that $S$ is the projection of a spectrahedron, i.e., it has asemidefinite representation. The semidefinite representation used by Helton andNie arises from glueing together Lasserre relaxations of many small piecesobtained in a non-constructive way. By refining and varying their approach, weshow that we can simply take a Lasserre relaxation of the original systemitself. Such a result was provided by Helton and Nie with much more machineryonly under very technical conditions and after changing the description of $S$.

On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets