Rank-one convexity

Summary

A function on $m \times n$ matrices is rank-one convex if it is convex along every line in a rank-one direction $a \otimes b$. It is the weakest of the convexity notions used in the vectorial calculus of variations. Polyconvexity implies quasiconvexity, which implies rank-one convexity, and each is implied by ordinary convexity. For $C^2$ integrands it is the Legendre–Hadamard condition at every matrix.