{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $C \\subseteq \\R^n$ be any subset.\nThe *dual cone* of $C$ is the set\n$$C^* = \\set{y \\in \\R^n : x \\cdot y \\ge 0 \\text{ for all } x \\in C},$$\nwhere $x \\cdot y$ is the [[dot product|0S52TNNTH9]].\n\n## Remarks\n\nThe set $C^*$ is an intersection of closed half-spaces through $0$, so it is a closed [[convex cone|6VE4KA12VD]] that contains $0$.\nIf $C$ is a closed convex cone, then $(C^*)^* = C$.\n\nExample: for $C = \\R^n_{\\ge 0}$, one has $C^* = \\R^n_{\\ge 0}$.\nIf $y$ has nonnegative coordinates, then $x \\cdot y \\ge 0$ for each $x \\in C$; conversely, taking $x = e_i$ gives $y_i \\ge 0$.\nA cone with $C^* = C$ is called self-dual; the cone of positive semidefinite matrices with the trace inner product is another self-dual cone.\n\nNon-example: the vector $y = (1, -1)$ is not in the dual of $\\R^2_{\\ge 0}$, since $(0, 1) \\cdot y = -1 \u003c 0$.\n\nThe *polar cone* is $C^\\circ = -C^* = \\set{y : x \\cdot y \\le 0 \\text{ for all } x \\in C}$.\nIn a general real inner product space, or with a pairing between a space and its dual, the same formula defines the dual cone.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "QFY1JR1TK4",
    "kind": "definition",
    "msc": [
      "52A20",
      "90C25"
    ],
    "name": "Dual cone",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dual_cone_and_polar_cone"
      }
    ],
    "tag": "1ZDAGGGRP4",
    "uses": [
      "0XPR1E85W0",
      "0S52TNNTH9",
      "6VE4KA12VD"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/1ZDAGGGRP4/",
  "links": [
    {
      "field": "defines",
      "target": "QFY1JR1TK4"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    },
    {
      "field": "uses",
      "target": "0S52TNNTH9"
    },
    {
      "field": "uses",
      "target": "6VE4KA12VD"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/1ZDAGGGRP4.md",
  "msc": [
    "52A20",
    "90C25"
  ],
  "name": "Dual cone",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1ZD/1ZDAGGGRP4.md",
  "status": null,
  "tag": "1ZDAGGGRP4",
  "uri": "m2://1ZDAGGGRP4",
  "url": "https://tamnd.github.io/math2/wiki/dual-cone-definition/",
  "writing": {
    "tag": "1ZDAGGGRP4",
    "kind": "definition",
    "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
      }
    ]
  }
}