By a classical estimate in function theory, each subharmonic function u(z) on the unit disk that is bounded above by 1 and bounded by 0 on the real axis must satisfy a bound of the form u(z) ≤ A|Imz| on smaller subdisks. When an analogous estimate holds for the plurisubharmonic functions in a neighborhood of a real point ξ in an analytic variety, the variety is said to satisfy the local Phragmén-Lindelöf condition at ξ. Interest in such conditions originated from a theorem of Hörmander who showed that the surjective constant coefficient linear partial differential operators on the space of real analytic functions on ℝ n are characterized in terms of these conditions. We give a new geometric condition on a local variety that is necessary in order that the local Phragmén-Lindelöf condition holds, and is sufficient in the case of varieties of dimension 1 or 2.