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 Q_(p)(K) - a more restrictive quasiconvex hull of K