On curves minimizing polyhedral functionals in \(R^ n\) (Q1814276)

If \(M\subset\mathbb{R}^ n\) is a \(k\)-dimensional submanifold and \(L\) a Lagrangian of degree \(k\) on \(\mathbb{R}^ n\), i.e. a continuous mapping from the Grassmannian \(G(n,k)\) into the positive reals, one considers the integral \(J(M)\) of \(L\) over \(M\) defined by \(J(M)=\int_ M L(T_ x M)dH^ k(x)\). Here, \(H^*\) denotes \(k\)-dimensional Hausdorff-measure. In particular, if \(L\equiv 1\) the integral \(J(M)\) equals the area \(H^ k(M)\), and the extremals are minimal surfaces. In this paper the author considers the case \(k=1\) and assumes that the Lagrangian is polyhedral, i.e. \(L\) is considered to be a norm on \(\mathbb{R}^ n\) such that the corresponding unit ball is a convex polyhedron. The main result is Theorem 2 which gives a necessary and sufficient condition for a piecewise differentiable curve to be minimizing w.r.t. the integral \(J\). Unfortunately the author does not mention the deep and important work of Jean Taylor on crystalline integrands.