On rank one convex functions that are homogeneous of degree one (Q289903)

Denote by \(\mathbb R^{N\times n}\) the set of all \(N\times n\) matrices with real entries. A function \(f:\mathbb R^{N\times n}\to \mathbb R\) is called rank-one convex if \(f(t\xi+(1-t)\eta)\leq t f(\xi)+(1-t)f(\eta)\) for all \(\xi,\eta\in \mathbb R^{N\times n}\) with rank\((\xi-\eta)\leq 1\). A cone of directions in a real vector space \(\mathcal V\) is a subset \(\mathcal D\) of \(\mathcal V\) that spans \(\mathcal V\) and such that \(tx\in\mathcal D\) for all \(x\in \mathcal D\) and \(t\in\mathbb R\). A subset \(\mathcal A\) of \(\mathcal V\) is called \(\mathcal D\)-convex if \([x,y]\subset \mathcal A\) for all \(x,y\in \mathcal A\) with \(x-y\in D\). A function \(f:\mathcal S\to\overline {\mathbb R} \, (=[-\infty,\infty])\) is called weakly \(\mathcal D\)-convex if \(f|_{[x,y]}\) is convex for all \(x,y\in \mathcal S\) such that \(x-y\in D\) and \([x,y]\subset \mathcal S\), and \(\mathcal D\)-convex if its extension \(F\) to \(\mathcal V\), taking the value \(+\infty\) on \(\mathcal V\setminus \mathcal S\), is weakly \(\mathcal D\)-convex. The main result of the paper (Theorem 1.1) asserts that any \(\mathcal D\)-convex and positively homogeneous function \(f:\mathcal C\to\mathbb R\) defined on an open convex cone \(\mathcal C\) in a finite dimensional normed space \(\mathcal V\) is convex at each point of \(\mathcal D\cap\mathcal C\). More exactly, for every \(x_0\in \mathcal D\cap\mathcal C\) there exists a linear functional \(\ell: \mathcal V\to\mathbb R\) such that \(\ell\leq f\) and \(\ell(x_0)=f(x_0)\). In particular, every rank-one convex and positively 1-homogeneous function \(f:\mathbb R^{N\times n}\to \mathbb R\) is convex at each point of the rank-one cone \(\{\xi \in \mathbb R^{N\times n}:\) rank \(\xi \leq 1\}\). Among the consequences of this abstract result one can mention various generalizations of the Ornstein \(L^1\) non-inequalities, see [\textit{O. Ornstein}, Arch. Ration. Mech. Anal. 11, 40--49 (1962; Zbl 0106.29602)]. Applications to gradient Young measures are considered as well.