  We give a scenario-based treatment plan optimization formulation that is
equivalent to planning with geometric margins if the scenario doses are
calculated using the static dose cloud approximation. If the scenario doses are
instead calculated more accurately, then our formulation provides a novel
robust planning method that overcomes many of the difficulties associated with
previous scenario-based robust planning methods. In particular, our method
protects only against uncertainties that can occur in practice, it gives a
sharp dose fall-off outside high dose regions, and it avoids underdosage of the
target in ``easy'' scenarios. The method shares the benefits of the previous
scenario-based robust planning methods over geometric margins for applications
where the static dose cloud approximation is inaccurate, such as irradiation
with few fields and irradiation with ion beams. These properties are
demonstrated on a suite of phantom cases planned for treatment with scanned
proton beams subject to systematic setup uncertainty.
