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  "body": "\n## Definition\n\nLet $m, n \\ge 1$ and let $f \\colon \\R^{m \\times n} \\to \\R$ be continuous.\nThe function $f$ is *quasiconvex* (in the sense of Morrey) if\n$$f(\\xi) \\le \\frac{1}{\\abs{D}} \\int_D f\\big(\\xi + \\nabla \\varphi(x)\\big) \\, dx$$\nfor every bounded open set $D \\subseteq \\R^n$, every $\\xi \\in \\R^{m \\times n}$ and every $\\varphi \\in W^{1,\\infty}_0(D; \\R^m)$.\nHere $W^{1,\\infty}_0(D; \\R^m)$ is the [[Sobolev space|M4CEZ5K9HP]] of Lipschitz maps that vanish on $\\partial D$.\nThe number $\\abs{D}$ is the Lebesgue measure of $D$.\n\n## Remarks\n\nIf the inequality holds for one bounded open set $D$ with $\\abs{\\partial D} = 0$, it holds for all of them; often $D$ is the unit cube.\n\nExample: every [[convex|07GR3TNW4H]] $f$ is quasiconvex.\nSince $\\varphi = 0$ on $\\partial D$, the average of $\\nabla \\varphi$ over $D$ is $0$, and Jensen's inequality gives the inequality.\n\nNon-example: $f(\\xi) = -\\abs{\\xi}^2$ is not quasiconvex.\nFor $\\varphi \\ne 0$, the cross term $2 \\int_D \\xi : \\nabla \\varphi \\, dx$ vanishes, so the average of $f(\\xi + \\nabla \\varphi)$ is $-\\abs{\\xi}^2 - \\frac{1}{\\abs{D}} \\int_D \\abs{\\nabla \\varphi}^2 \\, dx \u003c f(\\xi)$.\n\nEvery quasiconvex function is [[rank-one convex|GK4P8G0PB2]].\nFor $m = 1$ or $n = 1$, quasiconvexity is ordinary convexity.\nFor $m \\ge 3$ and $n \\ge 2$, Šverák gave a rank-one convex function that is not quasiconvex; the case $m = 2$ is open.\n",
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