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  "body": "\n## Summary\n\nA function $f$ on $m \\times n$ matrices is quasiconvex in the sense of Morrey if perturbations with zero boundary values never help.\nThat is, the average of $f(\\xi + \\nabla \\varphi)$ is never below $f(\\xi)$.\nUnder suitable growth conditions on $f$, it is exactly the condition under which $\\int_\\Omega f(\\nabla u) \\, dx$ is weakly lower semicontinuous on Sobolev spaces of vector-valued maps.\nIt lies between polyconvexity and [[rank-one convexity|GK4P8G0PB2]].\nIt coincides with convexity when $m = 1$ or $n = 1$.\nIt is a different notion from the [[quasiconvex functions|S4RMH7JQB8]] of optimisation, which are defined by convex sublevel sets.\n\n## Explanations\n\n### L1 human\n\nTake a linear map with constant gradient $\\xi$ on a region, and wiggle it inside while keeping its boundary values.\nQuasiconvexity says that the wiggled map always has at least as much energy as the linear one.\nSo, on average, an affine map is the cheapest way to achieve given affine boundary values.\nConvex functions have this property by Jensen's inequality, because the average of $\\nabla \\varphi$ over the region is zero.\nThe condition is hard to check in general, because it is not local in $\\xi$: it involves all test maps $\\varphi$.\n",
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