Some optimal embeddings of symplectic ellipsoids

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.