  Gelfand-Tsetlin polytopes are classical objects in algebraic combinatorics
arising in the representation theory of $\mathfrak{gl}_n(\mathbb{C})$. The
integer point transform of the Gelfand-Tsetlin polytope $\mathrm{GT}(\lambda)$
projects to the Schur function $s_{\lambda}$. Schur functions form a
distinguished basis of the ring of symmetric functions; they are also special
cases of Schubert polynomials $\mathfrak{S}_{w}$ corresponding to Grassmannian
permutations.
  For any permutation $w \in S_n$ with column-convex Rothe diagram, we
construct a polytope $\mathcal{P}_{w}$ whose integer point transform projects
to the Schubert polynomial $\mathfrak{S}_{w}$. Such a construction has been
sought after at least since the construction of twisted cubes by Grossberg and
Karshon in 1994, whose integer point transforms project to Schubert polynomials
$\mathfrak{S}_{w}$ for all $w \in S_n$. However, twisted cubes are not honest
polytopes; rather one can think of them as signed polytopal complexes. Our
polytope $\mathcal{P}_{w}$ is a convex polytope. We also show that
$\mathcal{P}_{w}$ is a Minkowski sum of Gelfand-Tsetlin polytopes of varying
sizes. When the permutation $w$ is Grassmannian, the Gelfand-Tsetlin polytope
is recovered. We conclude by showing that the Gelfand-Tsetlin polytope is a
flow polytope.
