Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices

Abstract

Continuing earlier work by Szekelyhidi, we describe the topological and geometric structure of so-called T 4 -configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of T 4 -configurations in $\mathbb{R}^{2\times 2}$ is very rich; in particular, their collection is open as a subset of $(\mathbb{R}^{2\times 2})^{4}$ . Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect T 4 -configurations.