The Euclidean space has a nice property: every closed convex subset is the image of the whole space under a map that is simultaneously:
- a contraction, meaning for all ;
- a retraction, meaning for all .
Indeed, we can simply define to be the (unique) nearest point of to ; it takes a bit of work to verify that is a contraction, but not much.
In other normed spaces, this nearest point projection does not work that well. For example, take , the two-dimensional space with the Manhattan metric. Consider the line which is a closed and convex set. The nearest point of to is : moving straight up, since changing the first coordinate doesn’t pay off. Since remains fixed, the nearest point projection increases some pairwise distances, in this case from to .
However, there is a contractive retraction onto this line, given by the formuls . Indeed, this is a linear map that fixes the line pointwise and has norm because
More generally, in every normed plane, every closed convex subset admits a contractive retraction. To prove this, it suffices to consider closed halfplanes, since a closed convex set is an intersection of such, and contractions form a semigroup. Furthermore, it suffices to consider lines, because having a contractive retraction onto a line, we can redefine it to be the identity map on one side of the line, and get a contractive retraction onto a halfplane.
Such a retraction onto a line, which is a linear map, is illustrated below.
Given the unit circle (black) and a line (blue), draw supporting lines (red) to the unit circle at the points where it meets the line. Project onto the blue line along the red ones. By construction, the image of the unit disk under this projection is contained in the unit disk. This precisely means that the map has operator norm .
In spaces of dimensions or higher, there are closed convex subsets without a contractive retraction. For example, consider the plane in passing through the points , , and . This plane has equation . The point is at distance from each of A,B,C, and it does not lie on the plane. For any point E other than D, at least one of the distances AE, BE, CE exceeds 1. More precisely, the best place to project D is which is at distance from A, B, and C.
Two natural questions: (a) is there a nice characterization of normed spaces that admit a contractive retraction onto every closed convex subset? (b) what is the smallest constant such that every -dimensional normed space admits an -Lipschitz retraction onto every closed convex subset?
(The answers may well be known, but not to me at present.)