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We construct symplectic embeddings in dimension    2  n  ⩾  6    of ellipsoids into the product of a   4  ‐ball or four‐dimensional cube with Euclidean space. A sequence of the embeddings is seen to be optimal by a quantitative version of the embedding obstructions introduced in Hind and Kerman [ Invent. Math . 196 (2014) 383–452].  In the limiting case when our ellipsoids approach a cylinder, we recover an embedding of Guth [ Invent. Math . 172 (2008) 477–489]. However, for compact ellipsoids our embedding gives sharper results. At the other end of the scale, for certain convergent sequences of ellipsoids we reproduce estimates given by stabilizing four‐dimensional embeddings of McDuff and Schlenk [ Ann. of Math . 175 (2012) 1191–1282], in the ball case and Frenkel and Müller [ J. Symplectic Geom ., to appear], in the cube case.

Some optimal embeddings of symplectic ellipsoids