Symplectic embeddings of the $ell_p$-sum of two discs

In this paper we study symplectic embedding questions for the $ell_p$-sum of two discs in ${mathbb R}^4$, when $1 leq p leq infty$. In particular, we compute the symplectic inner and outer radii in these cases, and show how different kinds of embedding rigidity and flexibility phenomena appear as a function of the parameter $p$.