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  "body": "\n## Definition\n\nFix $n \\ge 1$.\nLet $B^{2n}(1)$ be the open unit ball and $Z^{2n}(1) = \\set{(x, y) \\in \\R^{2n} : x_1^2 + y_1^2 \u003c 1}$ the open unit cylinder, both with the standard symplectic form $\\omega_0$.\nA *symplectic capacity* on [[symplectic manifolds|CAN490EPRT]] of dimension $2n$ is a function $c$ assigning to each such manifold $(M, \\omega)$ a value $c(M, \\omega) \\in [0, \\infty]$ such that:\n\n1. (Monotonicity) If there is a symplectic embedding $(M, \\omega) \\to (N, \\tau)$, then $c(M, \\omega) \\le c(N, \\tau)$.\n2. (Conformality) $c(M, \\lambda \\omega) = \\abs{\\lambda} \\, c(M, \\omega)$ for every real $\\lambda \\ne 0$.\n3. (Normalization) $c(B^{2n}(1), \\omega_0) = c(Z^{2n}(1), \\omega_0) = \\pi$.\n\n## Remarks\n\nExample: the *Gromov width* $w(M, \\omega) = \\sup \\set{\\pi r^2 : B^{2n}(r) \\text{ embeds symplectically in } (M, \\omega)}$.\nMonotonicity holds because an embedding into $M$ composed with $M \\to N$ is an embedding into $N$.\nConformality holds because scaling by $\\sqrt{\\abs{\\lambda}}$ (and swapping $x$ and $y$ if $\\lambda \u003c 0$) maps $(B(r), \\lambda\\omega_0)$ to $(B(\\sqrt{\\abs{\\lambda}}\\, r), \\omega_0)$ symplectically.\nThe value $\\pi$ on the ball is clear, and the value $\\pi$ on the cylinder is exactly [[Gromov's non-squeezing theorem|RDX55HQ3XT]].\n\nNon-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.\n\nSome authors drop the normalization and only ask for nontriviality, $0 \u003c c(B^{2n}(1))$ and $c(Z^{2n}(1)) \u003c \\infty$.\nOther examples are the Hofer–Zehnder capacity and the Ekeland–Hofer capacities.\nThe Gromov width is the smallest normalized capacity.\n",
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