On Rank One Convex Functions that are Homogeneous of Degree One

We show that positively 1-homogeneous rank one convex functions are convex at 0 and at matrices of rank one. The result is a special case of an abstract convexity result that we establish for positively 1-homogeneous directionally convex functions defined on an open convex cone in a finite dimensional vector space. From these results we derive a number of consequences including various generalizations of the Ornstein L-1 non inequalities. Most of the results were announced in (C R Acad Sci Paris Ser I 349:407-409, 2011).