  The symplectic cone of a closed oriented 4-manifold is the set of cohomology
classes represented by symplectic forms. A well-known conjecture describes this
cone for every minimal Kaehler surface. We consider the case of the elliptic
surfaces E(n) and focus on a slightly weaker conjecture for the closure of the
symplectic cone. We prove this conjecture in the case of the spin surfaces
E(2m) using inflation and the action of self-diffeomorphisms of the elliptic
surface. An additional obstruction appears in the non-spin case.
