{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "6K8GPMHWBB"
    },
    {
      "field": "uses",
      "target": "TX0Z2VVKH3"
    },
    {
      "field": "uses",
      "target": "YV9QXZA4YG"
    },
    {
      "field": "defines",
      "target": "MVYTPJ3V05"
    }
  ],
  "body": "\n## Summary\n\nA set of capacity zero is a set so thin that no unit charge of finite energy can be placed on it.\nIn the plane the relevant capacity is the [[logarithmic capacity|DBQXRTC0VN]], and in $\\R^n$ with $n \\ge 3$ it is the Newtonian [[capacity|4YZVFGJVSZ]].\nFor Borel sets, having capacity zero is the same as being a [[polar set|C95QJ7YH5T]].\nThese sets are the exceptional sets of potential theory: theorems often hold everywhere except on a set of capacity zero.\n\n## Explanations\n\n### L1 human\n\nVolume and area cannot tell apart a point, a curve and a surface inside space: all have volume zero.\nCapacity is finer.\nIn space a point and a straight segment have capacity zero, but a flat disc does not.\nIn the plane a point and a countable set have capacity zero, but a segment does not.\nSets of capacity zero are invisible to harmonic functions that stay bounded.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "31B15",
      "31A15",
      "31C15"
    ],
    "name": "Sets of capacity zero",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Capacity_of_a_set"
      }
    ],
    "tag": "W9N1TR1X8P"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/W9N1TR1X8P/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/W9N1TR1X8P.md",
  "msc": [
    "31B15",
    "31A15",
    "31C15"
  ],
  "name": "Sets of capacity zero",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/W9N/W9N1TR1X8P.md",
  "status": null,
  "tag": "W9N1TR1X8P",
  "uri": "m2://W9N1TR1X8P",
  "url": "https://tamnd.github.io/math2/wiki/sets-of-capacity-zero/",
  "writing": {
    "tag": "W9N1TR1X8P",
    "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
      }
    ]
  }
}