  An influential result of McDuff and Schlenk asserts that the function that
encodes when a four-dimensional symplectic ellipsoid can be embedded into a
four-dimensional ball has a remarkable structure: the function has infinitely
many corners, determined by the odd-index Fibonacci numbers, that fit together
to form an infinite staircase.
  This work has recently led to considerable interest in understanding when the
ellipsoid embedding function for other symplectic 4-manifolds is partly
described by an infinite staircase. We provide a general framework for
analyzing this question for a large family of targets, called finite type
convex toric domains, which we prove generalizes the class of closed toric
symplectic 4-manifolds. When the target is of finite type, we prove that any
infinite staircase must have a unique accumulation point a_0, given as the
solution to an explicit quadratic equation. Moreover, we prove that the
embedding function at a_0 must be equal to the classical volume lower bound. In
particular, our result gives an obstruction to the existence of infinite
staircases that we show is powerful.
  In the special case of rational convex toric domains, we can say more. We
conjecture a complete answer to the question of existence of infinite
staircases, in terms of six families that are distinguished by the fact that
their moment polygon is reflexive. We then provide a uniform proof of the
existence of infinite staircases for our six families, using two tools. For the
first, we use recursive families of almost toric fibrations to find symplectic
embeddings. For the second tool, we find recursive families of convex lattice
paths that provide obstructions to embeddings. We conclude by reducing our
conjecture that these are the only infinite staircases among rational convex
toric domains to a question in number theory related to a classic work of Hardy
and Littlewood.
