Symplectic Non-Convexity of Toric Domains

The paper proves Theorem 1.1—given any star-shaped toric domain X_Ω ⊂ R^4, there exist arbitrarily small (in volume) star-shaped perturbations X_{Ω̂} that are not symplectically convex—by a robust invariant-based method. It uses the Chaidez–Edtmair necessary condition c < ru · sys < C for symplectic convexity, shows ru and sys are symplectomorphism invariants, computes Ru(X_Ω) explicitly, and gives two explicit perturbations (“strangulation” and “strain”) that drive ru · sys below c or above C while keeping the volume change arbitrarily small, completing the proof of non-symplectic convexity (see Theorem 1.1 and the constructions in Section 4) . By contrast, the model’s argument hinges on a misapplication of moment-map convexity: it claims that for a compact, connected, Hamiltonian T^2-invariant set C, the moment image μ′(C) is convex. This is false in general; the Atiyah–Guillemin–Sternberg convexity theorem applies to the moment image of the entire compact Hamiltonian manifold, not to arbitrary invariant subsets. Indeed, for toric domains C = μ^{-1}(Ω), one can choose nonconvex Ω so that μ(C) = Ω is nonconvex, showing the claimed step cannot hold. The paper avoids this pitfall and proceeds via ru · sys, giving explicit, volume-controlled perturbations and bounds on these invariants .