On Some Quasiconvex Functions with Linear Growth

We establish (i) that the quasiconvexification of the distance function to any closed (possibly unbounded) subset of the space of conformal matrices $E_\partial$ in M²×² is bounded from below by the distance function itself, that is, Q dist(·, K) ≥ c dist(·, K), where c > 0 is a constant independent of K; (ii) some estimates of quasiconvexifications of the distance function to a closed subset of M²×² which is ‘supported’ by $E_\partial$; (iii) Q distᵖ(·, K) = Q distᵖ(·, Qₚ(K)) for any p ≥ 1 and any closed K ⊂ Mᴺ×ⁿ; (iv) for some nonconvex K ⊂ M²×², Q dist(·, K) is homogeneous of degree one, conjugate invariant and convex, and Q₁(K) = C(K).