A symplectic embedding of the cube with minimal sections and a question by Schlenk (Q6601760)

Symplectic embeddings play a relevant role in symplectic geometry; one important result is the non-squeezing theorem of \textit{M. Gromov} [Invent. Math. 82, 307--347 (1985; Zbl 0592.53025)] that states that the closed ball \(\bar{B}^{2n}_r\) does not symplectically embed into the closed unit symplectic cylinder \(\bar{B}^2_1 \times \mathbb{R}^{n-2}\), if \(r>1\). Here \(\bar{B}^m_r\) indicates the closed ball of radius \(r\) into the Euclidean space \(\mathbb{R}^m\). \textit{F. Schlenk} [Int. Math. Res. Not. 2003, No. 2, 77--107 (2003; Zbl 1038.53082)] has investigated how flexible symplectic embeddings are in the case \(r \leq 1\). In the present paper, the author proves that the open unit cube can be symplectically embedded into a longer polydisc in such a way that the area of each section satisfies a sharp bound and the complement of each section is path-connected; using this result, a variant of a question by \textit{F. Schlenk} [loc. cit.] is answered.\N\NFor the entire collection see [Zbl 1515.53004].