On Rank-One Convex Functions that are homogeneous of Degree One

TL;DR

This paper proves that positively 1-homogeneous rank-one convex functions are convex at specific points and extends these results through a broader convexity framework, leading to generalizations of Ornstein's inequalities.

Contribution

It establishes a new convexity property for positively 1-homogeneous rank-one convex functions and introduces an abstract convexity result for directionally convex functions.

Findings

Positively 1-homogeneous rank-one convex functions are convex at zero and rank-one matrices.

Derived generalizations of Ornstein's L^1 non-inequalities.

Established an abstract convexity theorem for directionally convex functions.

Abstract

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 $\LL^1$ non inequalities. Most of the results were announced in ({\em C.~R.~Acad.~Sci.~Paris, Ser.~I 349 (2011), 407--409}).