{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "9HHEJVCS2Q"
    },
    {
      "field": "specializes",
      "target": "Z7RVXBA2KB"
    },
    {
      "field": "defines",
      "target": "ZKG8D1S566"
    }
  ],
  "body": "\n## Summary\n\nA domain in $\\C^n$ with $C^2$ boundary is Levi pseudoconvex when the [[Levi form|80MTNXWW80]] of its boundary is positive semidefinite at every boundary point.\nThis is the differential form of [[pseudoconvexity|AHYZMSBBNH]]: for domains with $C^2$ boundary the two notions agree.\nIt gives a test of pseudoconvexity that can be checked by computing second derivatives.\n\n## Explanations\n\n### L1 human\n\nConvexity asks that the boundary never bends inwards in any real direction.\nLevi pseudoconvexity asks this only for the complex directions tangent to the boundary, and only for the part of the bending that holomorphic maps see.\nSo it is a much weaker condition than convexity, but it is the right one for holomorphic functions.\nFor example, the unit ball in $\\C^n$ is Levi pseudoconvex, and so is every convex domain with $C^2$ boundary.\nThe shell $\\set{1 \u003c \\abs{z} \u003c 2}$ in $\\C^2$ is not: along the inner sphere its boundary bends the wrong way in the complex tangent directions.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32T05"
    ],
    "name": "Levi pseudoconvex domain",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Pseudoconvexity"
      }
    ],
    "specializes": [
      "AHYZMSBBNH"
    ],
    "tag": "PETEQQCS0B"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/PETEQQCS0B/",
  "links": [
    {
      "field": "specializes",
      "target": "AHYZMSBBNH"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/PETEQQCS0B.md",
  "msc": [
    "32T05"
  ],
  "name": "Levi pseudoconvex domain",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PET/PETEQQCS0B.md",
  "status": null,
  "tag": "PETEQQCS0B",
  "uri": "m2://PETEQQCS0B",
  "url": "https://tamnd.github.io/math2/wiki/levi-pseudoconvex-domain/",
  "writing": {
    "tag": "PETEQQCS0B",
    "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
      }
    ]
  }
}