  The relationship between certain geometric objects called polytopes and
scattering amplitudes has revealed deep structures in QFTs. It has been
developed in great depth at the tree- and loop-level amplitudes in
$\mathcal{N}=4~\text{SYM}$ theory and has been extended to the scalar $\phi^3$
and $\phi^4$ theories at tree-level. In this paper, we use the generalized BCFW
recursion relations for massless planar $\phi^4$ theory to constrain the
weights of a class of geometric objects called Stokes polytopes, which manifest
in the geometric formulation of $\phi^4$ amplitudes. We see that the weights of
the Stokes polytopes are intricately tied to the boundary terms in $\phi^4$
theories. We compute the weights of $N=1,2$, and $3$ dimensional Stokes
polytopes corresponding to six-, eight- and ten-point amplitudes respectively.
We generalize our results to higher-point amplitudes and show that the
generalized BCFW recursions uniquely fix the weights for an $n$-point
amplitude.
