Nonlinear orthogonal projection

Abstract

We discuss some properties of an orthogonal projection onto a subset of a Euclidean space. The special stress is laid on projection's regularity and characterization of the interior of its domain. 0. Introduction. Let M be a non-empty subset of a metric space Z. We define a relation P ⊂ Z × M , which we call the orthogonal projection onto M . Its domain is dom P := {z ∈ Z : there exists a unique point z ′ ∈ M such that d(z, z ′) = ̺(z, M)} , where d denotes the metric of Z and ̺(z, M) := inf x∈M d(z, x). Obviously, M ⊂ dom P. The orthogonal projection of z ∈ dom P is defined to be the unique point (z ′ =) P(z) ∈ M which realizes the distance of z to M . If M is a closed linear subspace of a Hilbert space Z, then P is the well-known linear orthogonal projection: Z = dom P → M . The need of considering orthogonal projections onto non-linear sets has been noticed since a long time. For example, if Z = R n and M is a smooth (or analytic) submanifold, then the composition f • P| int dom P is the most natural smooth (analytic) extension of a given smooth (analytic) function f : M → R on an open neighbourhood of M (because in this case M ⊂ int dom P (see the generalization (3.8) of the classical result of Federer [5] and (4.1))). Of course, there are other methods of extending such functions, e.g. in the non-analytic case by local straightening of M or by applying Whitney's theory. However, in numerous problems the extension f • P is most useful, since it is simple and effective. The set int dom P is in some sense a star-shaped neighbourhood of M (see (1.5), (3.13)), so the retrac-tion P| int dom P is helpful in studies on differentiable homotopy, e.g. for a given solenoidal vector field v : G → R n vanishing on the boundary of a 1991 Mathematics Subject Classification: 58B10, 51Kxx.