  Fleischer and Mannel (FM) have shown that it may become possible to constrain
the angle $\gamma$ of the unitarity triangle from measurements of various
$B\to\pi K$ decays. This constraint is independent of hadronic uncertainties to
the few percent level. We show that, within the Standard Model, the FM bound
gives strong constraints on the CKM parameters. In particular, it could predict
a well defined sign for $\sin2\gamma$ and $\sin2\alpha$. In a class of
extensions of the Standard Model, where the New Physics affects only $\Delta
B=2$ (and, in particular, not $\Delta B=1$) processes, the FM bound can lead to
constraints on CP asymmetries in $B$ decays into final CP eigenstates even if
$B-\bar B$ mixing is dominated by unknown New Physics. In our analysis, we use
a new method to combine in a statistically meaningful way the various
measurements that involve CKM parameters.
