{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "EPN6C0X93V"
    }
  ],
  "body": "\n## Summary\n\nA symplectic capacity is a number attached to symplectic manifolds, or to subsets of $\\R^{2n}$, that measures their size.\nIt is compatible with symplectic embeddings.\nIt cannot grow under symplectic embeddings and is normalized so that a ball and a cylinder of the same radius have the same capacity.\nThe existence of capacities is equivalent in strength to [[Gromov's non-squeezing theorem|RDX55HQ3XT]], and capacities are a main tool of quantitative symplectic topology.\n\n## Explanations\n\n### L1 human\n\nVolume is a natural size for a region of space, but in symplectic geometry it misses much.\nA thin infinite cylinder has infinite volume, yet a large ball cannot be fitted into it symplectically.\nA capacity is a size that sees this: it behaves like an area, measured in a two-dimensional symplectic direction.\nThe simplest one is the Gromov width, the area $\\pi r^2$ of the largest ball of radius $r$ that fits into a set by a symplectic embedding.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "53D35",
      "57R17"
    ],
    "name": "Symplectic capacity",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Non-squeezing_theorem"
      }
    ],
    "tag": "8KCB5KJKNT"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/8KCB5KJKNT/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/8KCB5KJKNT.md",
  "msc": [
    "53D35",
    "57R17"
  ],
  "name": "Symplectic capacity",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/8KC/8KCB5KJKNT.md",
  "status": null,
  "tag": "8KCB5KJKNT",
  "uri": "m2://8KCB5KJKNT",
  "url": "https://math2.tamnd.com/wiki/symplectic-capacity/",
  "writing": {
    "tag": "8KCB5KJKNT",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}