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On various semiconvex relaxations of the squared-distance function

Published online by Cambridge University Press:  14 November 2011

K. Zhang
K. Zhang
Affiliation:
Department of Mathematics, Macquarie University, Sydney 2109, Australia (kewei@maths. ics.mq.edu.au)
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Extract

For the Euclidean squared-distance function f(·) = dist2(·, K), with K ⊂ MN×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 < Nn, K is convex if and only if f equals one of the semiconvex relaxations when dist2(P, K) is sufficiently large, and for case (i), P ∈MNxn; for case (ii), P ∈ Ek—a k-dimensional plane containing C(K). We also give some estimates of the difference between dist2(P, K) and its semiconvex relaxations. Some possible extensions to more general p-distance functions are also considered.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 6 , 1999 , pp. 1309 - 1323
Copyright
Copyright © Royal Society of Edinburgh 1999

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