ECH capacities give obstructions to symplectically embedding one symplecticfour-manifold with boundary into another. We compute the ECH capacities of alarge family of symplectic four-manifolds with boundary, called "concave toricdomains". Examples include the (nondisjoint) union of two ellipsoids in$\mathbb{R}^4$. We use these calculations to find sharp obstructions to certainsymplectic embeddings involving concave toric domains. For example: (1) wecalculate the Gromov width of every concave toric domain; (2) we show that manyinclusions of an ellipsoid into the union of an ellipsoid and a cylinder are"optimal"; and (3) we find a sharp obstruction to ball packings into certainunions of an ellipsoid and a cylinder.
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