On various semiconvex relaxations of the squared-distance function
Kewei Zhang · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 1999
For the Euclidean squared-distance function f(·) = dist 2 (·, K) , with K ⊂ M N×n , we show that K is convex if and only if f(·) equals either its rank-one convex, quasiconvex or polyconvex relaxations. We also establish that if (i) K is compact and contractible or (ii) dim C(K) = k < N n , K is convex if and only if f equals one of the semiconvex relaxations when dist 2 (P, K) is sufficiently large, and for case (i), P ∈M Nxn ; for case (ii), P ∈ E k —a k -dimensional plane containing C(K) . We also give some estimates of the difference between dist 2 (P, K) and its semiconvex relaxations. Some possible extensions to more general p -distance functions are also considered.