  The purpose of this paper is twofold. First, we provide an optimal
$\Omega(\sqrt{n})$ bits lower bound for any two-way protocol for the Vector in
Subspace Communication Problem which is of bounded total rank. This result
complements Raz's $O(\sqrt{n})$ protocol, which has a simple variant of bounded
total rank. Second, we present a plausible mathematical conjecture on a measure
concentration phenomenon that implies an $\Omega(\sqrt{n})$ lower bound for a
general protocol. We prove the conjecture for the subclass of sets that depend
only on $O(\sqrt{n})$ directions.
