Gromov width of Katok ellipsoids

Determine the Gromov width of the almost toric ellipsoids \(E_{2;1,1}(\alpha,\beta)\) and \(E_{2,1}(\alpha,\beta)\), equivalently the corresponding Katok ellipsoid domains associated with \(S^2\) and \(\mathbb{R}P^2\).

Background

The paper identifies $E_{2;1,1}(\alpha,\beta)$ with the unit codisc bundles of the Randers metrics inducing Katok’s examples on $S^2$, and identifies $E_{2,1}(\alpha,\beta)$ with the analogous domains associated with $\mathbb{R}P^2$. It computes their embedding regions into the relevant compact targets and establishes staircase structures.

The Gromov width is known in the round cases, namely for the ball-shaped domains $B_{2;1,1}(\lambda)$ and $B_{2,1}(\lambda)$, but the general two-parameter computation remains unresolved in the paper and is noted as a subject of forthcoming work.