Quasiconvexity in the calculus of variations is an integral inequality for a continuous function 
. The function
is quasiconvex if, for every 
and every smooth
function 
with compact support,
 |
where 
is the matrix of first partial
derivatives (Morrey 1952). The cube has unit volume,
so the integral is also the average value of the integrand.
This condition tests whether an affine deformation minimizes its integral against perturbations that vanish near the boundary.
It implies that 
is a rank-one convex function (Morrey
1952). This variational use of quasiconvexity differs from the convex
sublevel-set definition of a quasi-convex function.
