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On various semiconvex relaxations of the squared-distance function. (English) Zbl 0936.49009

Summary: For the Euclidean squared-distance function \(f(\cdot)= \text{dist}^2(\cdot, K)\), with \(K\subset M^{N\times n}\), we show that \(K\) is convex if and only if \(f(\cdot)\) equals either its rank-one convex, quasiconvex or polyconvex relaxations. We also establish that if (i) \(K\) is compact and contractible or (ii) \(\dim C(K)= k< Nn\), \(K\) is convex if and only if \(f\) equals one of the semiconvex relaxations when \(\text{dist}^2(P,K)\) is sufficiently large, and for case (i), \(P\in M^{N\times n}\); for case (ii), \(P\in E_k\) – a \(k\)-dimensional plane containing \(C(K)\). We also give some estimates of the difference between \(\text{dist}^2(P,K)\) and its semiconvex relaxations. Some possible extensions to more general \(p\)-distance functions are also considered.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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