Symplectic capacity

Remarks

Example: the Gromov width $w(M, \omega) = \sup \set{\pi r^2 : B^{2n}(r) \text{ embeds symplectically in } (M, \omega)}$. Monotonicity holds because an embedding into $M$ composed with $M \to N$ is an embedding into $N$. Conformality holds because scaling by $\sqrt{\abs{\lambda}}$ (and swapping $x$ and $y$ if $\lambda < 0$) maps $(B(r), \lambda\omega_0)$ to $(B(\sqrt{\abs{\lambda}}\, r), \omega_0)$ symplectically. The value $\pi$ on the ball is clear, and the value $\pi$ on the cylinder is exactly Gromov's non-squeezing theorem.

Non-example: for $n \ge 2$, the function $c(M, \omega) = \left( \int_M \omega^n / n! \right)^{1/n}$, suitably rescaled, satisfies monotonicity and conformality, but fails normalization, because the cylinder has infinite volume.

Some authors drop the normalization and only ask for nontriviality, $0 < c(B^{2n}(1))$ and $c(Z^{2n}(1)) < \infty$. Other examples are the Hofer–Zehnder capacity and the Ekeland–Hofer capacities. The Gromov width is the smallest normalized capacity.