With the wealth of new data from the B-factories, b -> d penguin decays become available for study, in addition to their b -> s counterparts that have proven an indespensable tool for the exploration of new-physics effects in flavour physics. A prominent example of the b -> d penguin transitions is $bar B^0_d to K^0 bar K^0$. We show that this decay can be charaterized in the Standard Model by a surface in the observable space of the direct and mixing-induced CP asymmetries and the branching ratio. The form of this surface, which is theoretically clean, implies a lower bound for the branching ratio that has recently been confirmed experimentally. If future measurements of the CP asymmetries yield a point away from the SM surface, this would be an interesting signal of new physics. We point out that the hadronic parameters in $bar B^0_d to K^0 bar K^0$ that parameterize the position on the SM surface are related to hadronic parameters in the B -> pi K system. The fact that the branching ratio of $bar B^0_d to K^0 bar K^0$ is very close to its lower bound yields interesting implications for B -> pi K even without knowledge of the CP asymmetries of $bar B^0_d to K^0 bar K^0$. The mechanism that produces the lower bound for $bar B^0_d to K^0 bar K^0$ is actually much more general; we derive lower bounds for various other b -> d penguin-induced processes, including B -> rho gamma and $B^pm to K^{(ast)pm} K^{(ast)}$. Some of these theoretical lower bounds are very close to the current experimental upper bounds.
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