  We study remarkable connections between twistor-string formulas for tree
amplitudes in ${\cal N}=4$ SYM and ${\cal N}=6$ ABJM, and the corresponding
momentum amplituhedron in the kinematic space of $D=4$ and $D=3$, respectively.
Based on the Veronese map to positive Grassmannians, we define a twistor-string
map from $G_{+}(2,n)$ to a $(2n{-}4)$-dimensional subspace of the 4d kinematic
space where the momentum amplituhedron of SYM lives. We provide strong evidence
that the twistor-string map is a diffeomorphism from $G_+(2,n)$ to the interior
of momentum amplituhedron; the canonical form of the latter, which is known to
give tree amplitudes of SYM, can be obtained as pushforward of that of former.
We then move to three dimensions: based on Veronese map to orthogonal positive
Grassmannian, we propose a similar twistor-string map from the moduli space
${\cal M}_{0,n}^+$ to a $(n{-}3)$-dimensional subspace of 3d kinematic space.
The image gives a new positive geometry which conjecturally serves as the
momentum amplituhedron for ABJM; its canonical form gives the tree amplitude
with reduced supersymmetries in the theory. We also show how boundaries of
compactified ${\cal M}_{0,n}^+$ map to boundaries of momentum amplituhedra for
SYM and ABJM corresponding to factorization channels of amplitudes, and in
particular for ABJM case the map beautifully excludes all unwanted channels.
