Subspaces without rank-one matrices and quasiconvexity

I give a survey on linear subspaces of real matrices without rank-one connections (SNR1) and their applications to quasiconvexity. I will discuss (i) a simple case of a two point set without rank-one connections, recalling the compactness result of Ball-James and the quasiconvex envelope of Kohn and Pipkin; (ii) some direct applications and results motivated from (i); (iii) some algebraic, topological and analytic properties of SNR1; (iv) applications to the quasiconvex hull, the unstable solution for a two-point problem and the separation of gradient Young measures.