Positive geometry for graviton scattering amplitudes
In plain words
Gravity amplitudes lack the cyclic ordering of particles that the gauge-theory shapes rely on. No shape whose volume-like form gives graviton scattering is known.
Precise statement
Find a positive geometry whose canonical form equals tree-level n-graviton amplitudes of Einstein gravity or $N=8$ supergravity in $D = 4$ for all helicity sectors, or the loop integrands of $N=8$ supergravity; or prove that gravity amplitudes cannot be canonical forms in the standard sense.
What would settle it
An explicit geometry reproducing all tree-level graviton amplitudes, or a proof of impossibility.
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