In this note we show that one open 4-dimensional ellipsoid embeds symplectically into another if and only if the ECH capacities of the first are no larger than those of the second. This proves a conjecture due to Hofer. The argument uses the equivalence of the ellipsoidal embedding problem with a ball embedding problem that was recently established by McDuff. Its method is inspired by Hutchings’ recent results on embedded contact homology (ECH) capacities but does not use them.
"The Hofer Conjecture on Embedding Symplectic Ellipsoids." J. Differential Geom. 88 (3) 519 - 532, July 2011. https://doi.org/10.4310/jdg/1321366358