  We present a convex geometry perspective to the Effective Field Theory (EFT)
parameter space. We show that the second $s$ derivatives of the forward EFT
amplitudes form a convex cone, whose extremal rays are closely connected with
states in the UV theory. For tree-level UV-completions, these rays are simply
theories with all UV particles living in at most one irreducible representation
of the symmetries of the theory. In addition, all the extremal rays are
determined by the symmetries and can be systematically identified via group
theoretical considerations. The implications are twofold. First, geometric
information encoded in the EFT space can help reconstruct the UV-completion. In
particular, we will show that the dim-8 operators are important in
reverse-engineering the UV physics from the Standard Model EFT, and thus
deserve more theoretical and experimental investigations. Second, theoretical
bounds on the Wilson coefficients can be obtained by identifying the boundaries
of the cone and are in general stronger than the current positivity bounds. We
show explicit examples of these new bounds, and demonstrate that they originate
from the scattering amplitudes corresponding to entangled states.
