Let $M(n,m)$ denote the real $m\times n$ matrices. A continuous function $f\colon M(n,m)\to \R$ is called {\it Morrey quasi-convex} if $$\int_{\R^n}(f(A+\nabla\vf(x))-f(A))\,dx\ge 0$$ for each smooth, compactly supported $\vf\colon\R^n\to\R^m$ and each $A\in M(n,m)$. The inequality can be interpreted as a stability requirement for variational integrals $\int_\Om f(\nabla u(x))\,dx$, and with some natural assumptions it is equivalent to the weak lower semi-continuity of the corresponding functional, as proved by Ch.\ B.\ Morrey in the early 1950s. A simple necessary condition for the quasi-convexity is that $f$ be convex along lines $t\to A+tB$ with $A,B\in M(n,m)$ and ${\rm rank\,} B=1$. For smooth functions $f$, the strict form of this {\it rank-one convexity} condition corresponds to the classical ellipticity  (the {\it Legendre-Hadamard condition}) for the associated Euler-Lagrange equation. An easy sufficient condition for the quasi-convexity is {\it polyconvexity} --- the possibility of expressing $f(X)$ as a convex function of the minors of $X$. For $m=1$ or $n=1$ all these notions reduce to standard convexity, but when $n,m\ge 2$ the relations between the various notions of convexity become interesting. 

It is known that the rank-one convexity of $f$ is not sufficient for the quasi-convexity when $n\ge 2$ and $m\ge 3$, but the situation with $n\ge 2$ and $m=2$ remains open. It was observed by several authors that the problem of optimal constants in the $L^p$ estimates for the complex Hilbert transform (with $p>1$) is a special case. The geometry related to the various convexity notions can also be used for constructing examples relevant to the regularity theory of the variational integrals above.

This talk will discuss some of these topics, including possible linear programming experiments aimed at investigating the validity of some of the unresolved inequalities.