  We study the log-rank conjecture from the perspective of point-hyperplane
incidence geometry. We formulate the following conjecture: Given a point set in
$\mathbb{R}^d$ that is covered by constant-sized sets of parallel hyperplanes,
there exists an affine subspace that accounts for a large (i.e.,
$2^{-{\operatorname{polylog}(d)}}$) fraction of the incidences. Alternatively,
our conjecture may be interpreted linear-algebraically as follows: Any rank-$d$
matrix containing at most $O(1)$ distinct entries in each column contains a
submatrix of fractional size $2^{-{\operatorname{polylog}(d)}}$, in which each
column contains one distinct entry. We prove that our conjecture is equivalent
to the log-rank conjecture.
  Motivated by the connections above, we revisit well-studied questions in
point-hyperplane incidence geometry without structural assumptions (i.e., the
existence of partitions). We give an elementary argument for the existence of
complete bipartite subgraphs of density $\Omega(\epsilon^{2d}/d)$ in any
$d$-dimensional configuration with incidence density $\epsilon$. We also
improve an upper-bound construction of Apfelbaum and Sharir (SIAM J. Discrete
Math. '07), yielding a configuration whose complete bipartite subgraphs are
exponentially small and whose incidence density is $\Omega(1/\sqrt d)$.
Finally, we discuss various constructions (due to others) which yield
configurations with incidence density $\Omega(1)$ and bipartite subgraph
density $2^{-\Omega(\sqrt d)}$.
  Our framework and results may help shed light on the difficulty of improving
Lovett's $\tilde{O}(\sqrt{\operatorname{rank}(f)})$ bound (J. ACM '16) for the
log-rank conjecture; in particular, any improvement on this bound would imply
the first bipartite subgraph size bounds for parallel $3$-partitioned
configurations which beat our generic bounds for unstructured configurations.
