  We consider the problem of characterizing the convex hull of the graph of a
bilinear function $f$ on the $n$-dimensional unit cube $[0,1]^n$. Extended
formulations for this convex hull are obtained by taking subsets of the facets
of the Boolean Quadric Polytope (BQP). Extending existing results, we propose a
systematic study of properties of $f$ that guarantee that certain classes of
BQP facets are sufficient for an extended formulation. We use a modification of
Zuckerberg's geometric method for proving convex hull characterizations
[Geometric proofs for convex hull defining formulations, Operations Research
Letters \textbf{44} (2016), 625--629] to prove some initial results in this
direction. In particular, we provide small-sized extended formulations for
bilinear functions whose corresponding graph is either a cycle with arbitrary
edge weights or a clique or an almost clique with unit edge weights.
