  We use positive S^1-equivariant symplectic homology to define a sequence of
symplectic capacities c_k for star-shaped domains in R^{2n}. These capacities
are conjecturally equal to the Ekeland-Hofer capacities, but they satisfy
axioms which allow them to be computed in many more examples. In particular, we
give combinatorial formulas for the capacities c_k of any "convex toric domain"
or "concave toric domain". As an application, we determine optimal symplectic
embeddings of a cube into any convex or concave toric domain. We also extend
the capacities c_k to functions of Liouville domains which are almost but not
quite symplectic capacities.
