\leq 1\right\}$$ while the standard $4$-dimensional polydisk $P(s,t)$ is given by $$P(s ,t) :=
  \left\{ z \in {\mathbb{C}}^2 ~|~ \pi|z_1|^2 \leq s,~ \pi|z_2|^2\leq t \right\}$$ Recently D. Muller [@Mu] showed that the problem of embedding an ellipsoid into a polydisc is equivalent with embedding a collection of balls of various sizes into the polydisc. Using this, we can state

\[muler\] Let $k$ be any integer greater than $8$ and let $a,s,t$ be any positive real numbers with $s < t.$ Denote by $\mu = \frac{a}{s}$. The following are equivalent:

1.  $E(a, k a) \hookrightarrow P(s,t)$

2.  If $k =2p+1 $ then $\frac{a}{s} \leq {\mathsf{w}}_k =\min\{1, c_{\operatorname{vol}},
        \frac{\mu + p}{2p+1} \}.$ If $k=2p$ then $\frac {a}{s} \leq {\mathsf{w}}_{k}
        =\min_{n \in {\mathbb{N}}} \{1, c_{\operatorname{vol}}, w_n \}.$ Moreover, there is an $n \in {\mathbb{N}}\cup \{\infty \}$ so that $\mu=\frac{a}{s} \in I_n$ and the precise value of this ${\mathsf{w}}_k$ is given by Theorem \[thm:MainTheoremTrivial\] part (ii).

The proof of that corollary is given at the end of the paper. In that section, we also give an alternative proof, valid only for the case when $b/a$ is an integer, of a recent result of M. Hutchings [@H2], [@H4] stating that the embedded contact homology (ECH) 