  We introduce tools from discrete convexity theory and polyhedral geometry
into the theory of West's stack-sorting map $s$. Associated to each permutation
$\pi$ is a particular set $\mathcal V(\pi)$ of integer compositions that
appears in a formula for the fertility of $\pi$, which is defined to be
$|s^{-1}(\pi)|$. These compositions also feature prominently in more general
formulas involving families of colored binary plane trees called troupes and in
a formula that converts from free to classical cumulants in noncommutative
probability theory. We show that $\mathcal V(\pi)$ is a transversal discrete
polymatroid when it is nonempty. We define the fertilitope of $\pi$ to be the
convex hull of $\mathcal V(\pi)$, and we prove a surprisingly simple
characterization of fertilitopes as nestohedra arising from full binary plane
trees. Using known facts about nestohedra, we provide a procedure for
describing the structure of the fertilitope of $\pi$ directly from $\pi$ using
Bousquet-M\'elou's notion of the canonical tree of $\pi$. As a byproduct, we
obtain a new combinatorial cumulant conversion formula in terms of
generalizations of canonical trees that we call quasicanonical trees. We also
apply our results on fertilitopes to study combinatorial properties of the
stack-sorting map. In particular, we show that the set of fertility numbers has
density $1$, and we determine all infertility numbers of size at most $126$.
Finally, we reformulate the conjecture that $\sum_{\sigma\in
s^{-1}(\pi)}x^{\text{des}(\sigma)+1}$ is always real-rooted in terms of
nestohedra, and we propose natural ways in which this new version of the
conjecture could be extended.
