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  "body": "\n## Summary\n\nA function on $m \\times n$ matrices is rank-one convex if it is convex along every line in a rank-one direction $a \\otimes b$.\nIt is the weakest of the convexity notions used in the vectorial calculus of variations.\nPolyconvexity implies quasiconvexity, which implies rank-one convexity, and each is implied by ordinary [[convexity|07GR3TNW4H]].\nFor $C^2$ integrands it is the Legendre–Hadamard condition at every matrix.\n\n## Explanations\n\n### L1 human\n\nTwo matrices that differ by a rank-one matrix can be the gradients of one continuous map on the two sides of a flat interface.\nFine layers of two such gradients form a laminate, and a laminate can mimic the average gradient.\nRank-one convexity says that such layering never lowers the energy below its value at the average.\nFor $2 \\times 2$ matrices the determinant is rank-one convex, in fact affine along rank-one lines, but it is not convex.\n",
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