The 2+1-convex hull of a~finite set

Abstract

Rank-one convexity is a weak form of convexity related to convex integration and the elusive notion of quasiconvexity, but more amenable both in theory and practice. However, exact algorithms for computing the rank one convex hull of a finite set are only known for some special cases of separate convexity with a finite number of directions. Both inner approximations either with laminates or T 4 {T_{4}} ’s and outer approximations through polyconvexity are known to be insufficient in general. We study ℝ 2 ⊕ ℝ {\mathbb{R}^{2}\oplus\mathbb{R}} -separately convex hulls of finite sets, which is a special case of rank-one convexity with infinitely many directions in which T 4 {T_{4}} ’s are known not to capture the rank one convex hull. When ℝ 3 {\mathbb{R}^{3}} is identified with a subset of 2 × 3 {2\times 3} matrices, it is known to correspond also to quasiconvexity. We propose new inner and outer approximations built upon systematic use of known results, and prove that they agree. The inner approximation allows to understand better the structure of the rank one convex hull. The outer approximation gives rise to a computational algorithm which, in some cases, computes the hull exactly, and in general builds a sequence that converges to the hull. We use and systematize all previous attempts at computing D-convex hulls, and bring new ideas that may help compute general D-convex hulls.