We re-examine the constraints imposed by causality and unitarity on the low-energy effective field theory expansion of four-particle scattering amplitudes, exposing a hidden “totally positive” structure strikingly similar to the positive geometries associated with grassmannians and amplituhedra. This forces the infinite tower of higher-dimension operators to lie inside a new geometry we call the “EFT-hedron”. We initiate a systematic investigation of the boundary structure of the EFT-hedron, giving infinitely many linear and non-linear inequalities that must be satisfied by the EFT expansion in any theory. We illustrate the EFT-hedron geometry and constraints in a wide variety of examples, including new consistency conditions on the scattering amplitudes of photons and gravitons in the real world.

The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less than 2 plays an important role in many problems in low-dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli space of convex hyperbolic and Euclidean polyhedra can be reduced to the study of deformations of cone-manifolds by doubling a polyhedron across its faces. This deformation theory has been understood by Hodgson and Kerckhoff [7] when the singular set has no vertices, and by Weiß [32] when the cone angles are less than . We prove here an infinitesimal rigidity result valid for cone angles less than 2, stating that infinitesimal deformations which leave the dihedral angles fixed are trivial in the hyperbolic case, and reduce to some simple deformations in the Euclidean case. The method is to treat this as a problem concerning the deformation theory of singular Einstein metrics, and to apply analytic methods about elliptic operators on stratified spaces. This work is an important ingredient in the local deformation theory of cone-manifolds by the second author [22]; see also the concurrent work by Weiß [33].

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We study different classes of Jensen measures for plurisubharmonic functions, in particular the relation between Jensen measures for continuous functions and Jensen measures for upper bounded functions. We prove an approximation theorem for plurisubharmonic functions in$B$-regular domain. This theorem implies that the two classes of Jensen measures coincide in$B$-regular domains. Conversely we show that if Jensen measures for continuous functions are the same as Jensen measures for upper bounded functions and the domain is hyperconvex, the domain satisfies the same approximation theorem as above.

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