  We consider operators acting on convex subsets of the unit hypercube. These
operators are used in constructing convex relaxations of combinatorial
optimization problems presented as a 0,1 integer programming problem or a 0,1
polynomial optimization problem. Our focus is mostly on operators that, when
expressed as a lift-and-project operator, involve the use of semidefiniteness
constraints in the lifted space, including operators due to Lasserre and
variants of the Sherali--Adams and Bienstock--Zuckerberg operators. We study
the performance of these semidefinite-optimization-based lift-and-project
operators on some elementary polytopes --- hypercubes that are chipped (at
least one vertex of the hypercube removed by intersection with a closed
halfspace) or cropped (all $2^n$ vertices of the hypercube removed by
intersection with $2^n$ closed halfspaces) to varying degrees of severity
$\rho$. We prove bounds on $\rho$ where these operators would perform badly on
the aforementioned examples. We also show that the integrality gap of the
chipped hypercube is invariant under the application of several
lift-and-project operators of varying strengths.
