Automatic convexity and Ornstein's L-one noninequalities

In  one-dimensional variational problems, ordinary convexity is the central notion as is characterizes existence of solutions (via the direct method), as well as their regularity. It is well known, since the pioneering work of Morrey, that the corresponding "quasi-convexity" notion in higher dimensions is of a less geometric character and much harder to understand. Since then a zoo of different (semi)-convexities approximating quasi-convexity was introduced. We present joint work with J.Kristensen (Oxford) which establishes a surprising coincidence of these notions with the class of 1-homogeneous functionals. Implications, among other things, is a unified approach to the Ornstein L^1-non-inequalities.