# Excursions into the calculus of variations and notions of convexity

A convex function is a function that lies above all its tangent (hyper)-planes. These types of functions are extremely useful in minimization problems as they have a single global minimum. However, they are also quite rare in practical applications. There exist several weaker notions of convexity which can better characterize non-convex functions. One of these notions, quasiconvexity, is equivalent to weak lower semi-continuity, an important property that implies the existence of a minimizing sequence's weak convergence to a minimum. However, quasiconvexity is difficult to characterize, but it implies another notion of convexity, rank-one convexity. If the reverse implication held as well, there would be an easier method to check weak lower semi-continuity as rank one convexity is easier characterized. However in 1952, Morrey conjectured that rank-one convexity does not imply quasiconvexity. In 1992, Sverak proved Morrey's conjecture, finding a counterexample that can be generalized to any function with domain of dimension $n \geq 2$ and codomain of $m \geq 3$. The case of $n = m = 2$ remains open, and was the target of our efforts this summer.