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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. [ABSTRACT FROM AUTHOR]