  Consider a finite system of non-strict polynomial inequalities with solution
set $S\subseteq\mathbb R^n$. Its Lasserre relaxation of degree $d$ is a certain
natural linear matrix inequality in the original variables and one additional
variable for each nonlinear monomial of degree at most $d$. It defines a
spectrahedron that projects down to a convex semialgebraic set containing $S$.
In the best case, the projection equals the convex hull of $S$. We show that
this is very often the case for sufficiently high $d$ if $S$ is compact and
"bulges outwards" on the boundary of its convex hull.
  Now let additionally a polynomial objective function $f$ be given, i.e.,
consider a polynomial optimization problem. Its Lasserre relaxation of degree
$d$ is now a semidefinite program. In the best case, the optimal values of the
polynomial optimization problem and its relaxation agree. We prove that this
often happens if $S$ is compact and $d$ exceeds some bound that depends on the
description of $S$ and certain characteristicae of $f$ like the mutual distance
of its global minimizers on $S$.
