Rank-One Connections at Infinity and Quasiconvex Hulls

We define p-rank-one connections at infinity for an unbounded set K in M^N×n and show that the quasiconvex hull Q_p(K) may be bigger than K if K has a p-rank-one connection, where Q_p(K) is the zero set of the quasiconvex relaxation of the p-distance function to K. We examine some examples and compare Q_p(K) with $\mathbf{Q}_p(K)$ - a more restrictive quasiconvex hull of K