Zhang, K. On various semiconvex relaxations of the squared-distance function. (English) Zbl 0936.49009 Proc. R. Soc. Edinb., Sect. A, Math. 129, No. 6, 1309-1323 (1999). 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. Cited in 6 Documents MSC: 49J45 Methods involving semicontinuity and convergence; relaxation 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:weak lower semicontinuity; integral functional; Euclidean squared-distance function; rank-one convex; quasiconvex; polyconvex; semiconvex; relaxations PDFBibTeX XMLCite \textit{K. Zhang}, Proc. R. Soc. Edinb., Sect. A, Math. 129, No. 6, 1309--1323 (1999; Zbl 0936.49009) Full Text: DOI References: [1] Zhang, Ann. Sci. Norm. Sup. Pisa Serie IV 19 pp 313– (1992) [2] Zhang, Proc. R. Soc. Edinb. A 127 pp 411– (1997) · Zbl 0883.49013 · doi:10.1017/S0308210500023726 [3] Šverák, In Micro structure and phase transition (1992) [4] Dacorogna, Direct methods in the calculus of variations (1989) · Zbl 0703.49001 · doi:10.1007/978-3-642-51440-1 [5] Dacorogna, Weak continuity and weak lower semicontinuity of nonlinear functionals 922 (1980) · Zbl 0676.46035 [6] DOI: 10.1007/BF00251759 · Zbl 0673.73012 · doi:10.1007/BF00251759 [7] Bhattacharya, Proc. R. Soc. Edinb. 124 pp 843– (1994) · Zbl 0808.73063 · doi:10.1017/S0308210500022381 [8] Ball, Proc. R. Soc. Edmb. A 114 pp 367– (1990) · Zbl 0716.49011 · doi:10.1017/S0308210500024483 [9] DOI: 10.1098/rsta.1992.0013 · Zbl 0758.73009 · doi:10.1098/rsta.1992.0013 [10] DOI: 10.1007/BF00279992 · Zbl 0368.73040 · doi:10.1007/BF00279992 [11] DOI: 10.1007/BF00275731 · Zbl 0565.49010 · doi:10.1007/BF00275731 [12] Rockafellar, Convex analysis (1970) · Zbl 0932.90001 · doi:10.1515/9781400873173 [13] Morrey, Multiple integrals in the calculus of variations (1966) · Zbl 0142.38701 [14] Mayer, Algebraic topology (1972) [15] Dret, C. R. Acad. Sci. Paris, Série I 318 pp 93– (1994) [16] Lay, Convex sets and their applications (1982) · Zbl 0492.52001 [17] DOI: 10.1016/S0294-1449(99)80006-7 · Zbl 0932.49015 · doi:10.1016/S0294-1449(99)80006-7 [18] DOI: 10.1002/cpa.3160390107 · Zbl 0609.49008 · doi:10.1002/cpa.3160390107 [19] DOI: 10.1007/BF00281246 · Zbl 0629.49020 · doi:10.1007/BF00281246 [20] DOI: 10.1007/BF01135336 · Zbl 0825.73029 · doi:10.1007/BF01135336 [21] DOI: 10.1007/BF00375279 · Zbl 0754.49020 · doi:10.1007/BF00375279 [22] Fonseca, J. Math. Pures Appl. 67 pp 175– (1988) [23] Zhang, J. Convex Analysis 5 pp 133– (1998) [24] DOI: 10.1016/S0294-1449(97)80132-1 · Zbl 0918.49014 · doi:10.1016/S0294-1449(97)80132-1 [25] Šverák, Proc. R. Soc. Edinb. A 120 pp 185– (1992) · Zbl 0777.49015 · doi:10.1017/S0308210500015080 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.