{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $m, n \\ge 1$ and let $f \\colon \\R^{m \\times n} \\to \\R$.\nFor $a \\in \\R^m$ and $b \\in \\R^n$, write $a \\otimes b$ for the $m \\times n$ matrix with entries $a_i b_\\alpha$; it has [[rank|0ZC9YTXZRY]] at most one.\nThe function $f$ is *rank-one convex* if for every $\\xi \\in \\R^{m \\times n}$, $a \\in \\R^m$ and $b \\in \\R^n$ the function\n$$t \\mapsto f(\\xi + t \\, a \\otimes b), \\quad t \\in \\R,$$\nis [[convex|07GR3TNW4H]].\n\n## Remarks\n\nExample: for $m = n = 2$, $f = \\det$ is rank-one convex.\nBy the matrix determinant lemma, $\\det(\\xi + t \\, a \\otimes b) = \\det \\xi + t \\, b^T \\operatorname{adj}(\\xi) \\, a$, which is affine in $t$.\nBut $\\det$ is not convex: along $\\xi(t) = \\operatorname{diag}(t, -t)$ it equals $-t^2$.\n\nNon-example: for $m = n = 1$, $f(\\xi) = -\\xi^2$ is not rank-one convex, since $t \\mapsto -t^2$ is not convex.\n\nFor $m = 1$ or $n = 1$ every matrix has rank at most one, so rank-one convexity is ordinary convexity.\nIf $f$ is $C^2$, it is rank-one convex exactly when $\\sum_{i,j,\\alpha,\\beta} \\frac{\\partial^2 f}{\\partial \\xi^i_\\alpha \\partial \\xi^j_\\beta}(\\xi) \\, a_i a_j b_\\alpha b_\\beta \\ge 0$ for all $\\xi$, $a$ and $b$, the Legendre–Hadamard condition.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "GK4P8G0PB2",
    "kind": "definition",
    "msc": [
      "49J45",
      "26B25"
    ],
    "name": "Rank-one convexity",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quasiconvexity_(calculus_of_variations)"
      }
    ],
    "tag": "TPC99RZ316",
    "uses": [
      "07GR3TNW4H",
      "0ZC9YTXZRY"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/TPC99RZ316/",
  "links": [
    {
      "field": "defines",
      "target": "GK4P8G0PB2"
    },
    {
      "field": "uses",
      "target": "07GR3TNW4H"
    },
    {
      "field": "uses",
      "target": "0ZC9YTXZRY"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/TPC99RZ316.md",
  "msc": [
    "49J45",
    "26B25"
  ],
  "name": "Rank-one convexity",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TPC/TPC99RZ316.md",
  "status": null,
  "tag": "TPC99RZ316",
  "uri": "m2://TPC99RZ316",
  "url": "https://tamnd.github.io/math2/wiki/rank-one-convexity-definition/",
  "writing": {
    "tag": "TPC99RZ316",
    "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
      }
    ]
  }
}