Abstract
We initiate a comprehensive investigation of the geometry of the ampli- tuhedron, a recently found geometric object whose volume calculates the integrand of scattering amplitudes in planar N = 4 SYM theory. We do so by introducing and studying its stratication, focusing on four-point amplitudes. The new stratication exhibits interesting combinatorial properties and positivity is neatly captured by per- mutations. As explicit examples, we nd all boundaries for the two and three loop amplitudes and related geometries. We recover the stratications of some of these ge- ometries from the singularities of the corresponding integrands, providing a non-trivial test of the amplituhedron/scattering amplitude correspondence. We nally introduce
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