  In this paper we focus on scattering amplitudes in maximally supersymmetric
Yang-Mills theory and define a long sought-after geometry, the loop momentum
amplituhedron, which we conjecture to encode tree and (the integrands of) loop
amplitudes in spinor helicity variables. Motivated by the structure of
amplitude singularities, we define an extended positive space, which enhances
the Grassmannian space featuring at tree level, and a map which associates to
each of its points tree-level kinematic variables and loop momenta. The image
of this map is the loop momentum amplituhedron. Importantly, our formulation
provides a global definition of the loop momenta. We conjecture that for all
multiplicities and helicity sectors, there exists a canonical logarithmic
differential form defined on this space, and provide its explicit form in a few
examples.
