Lee: If a space is an absolute -Lipschitz retract, with , can you say anything of its injective hull ? Answer: probably no. For instance, the injective hull of Euclidean is which seems very far from although is an absolute -Lipschitz retract.
1. Injective hulls: basics
1.1. Extremal functions
Let be a metric space. Leopoldo Nachbin, in the 1950’s, already considered the set
Definition 1 The minimal elements of are called extremal functions. The set of extremal functions is denoted by .
Note that if is compact, is extremal iff for all , there exists such that . In general,
is extremal iff for all , .
Note that all belong to . This yields a map .
Lemma 2 Extremal functions are -Lipschitz.
Proof: Given , define by and for . Then , so
Then the -norm defines a metric on .
Proposition 3 (Dress) There is a map such that
Proof: For , set , so is extremal iff . Note that . By definition, for all , ,
so , and . For every ,
Define (pointwise limit). Then . For all , , so
which tends to , so .
Remark 1 If , an infinite iteration is indeed needed to obtain .
Proposition 4 is injective.
Proof: Let be metric spaces, let be -Lipschitz. For , , put
Then . and is -Lipschitz. For , . We have
thus . Extend to by setting . Then the extended is -Lipschitz.
Theorem 5 (Isbell)
- is injective.
- If is -Lipschitz, and fixes pointwise, then is the identity.
- If is -Lipschitz and is an isometric embedding, then is an isometric embedding.
- If is an isometric embedding and is injective, then there is an isometric embedding such that .
Proof: 2. , so by minimality. 3. follows from 2. and 4. follows from 3.
2. First examples
2.1. Polyhedral structure on , finite
If is bounded, then . If is compact, so is (Arzela-Ascoli).
If is finite, is a polyhedron in . Note that and are convex, but is not in general. The polyhedral structure can be detected by looking at “equality graphs”. For , let be the set of pairs such that . Then is a graph with loops: iff . Also
iff has no isolated vertices.
Say a graph is admissible if there exists such that . Every admissible graph corresponds to a polyhedral cell in , whose interior points are precisely the functions such that . is a face of iff contains .
2.2. Dimension of
If two functions , have the same graph, then for all , , thus
Hence, if there is a path from to in of length , then
On connected components of containing an odd cycle, and coincide. On other (even) components, there is exactly one degree of freedom to vary without changing the graph.
Proposition 6 Define the rank as the number of even components. Then .
Example 1 .
Then the possible graphs are an edge (rank one) and one edge with one loop (rank zero). This produces an interval.
Example 2 .
Then is a tripod. Indeed, admissible graphs of extremal functions can have at most one even component. The complete graph is admissible but odd, producing a vertex. Two-edge admissible graphs contribute three segments. Hinges with a loop contribute three vertices.
The tripod lies in on a triangular face (standard simplex) at the bottom of polyhedron . It takes infinitely many steps (iterates of ) to map a constant function to . It is a bit stupid.
Lee: why don’t you consider the optimal such that belongs to ?
Example 3 has 4 points with one distance equal to and all other equal to or .
The graph with two disjoint edges (one of length ) is admissible and has rank , it contributes a -cell, a square sitting diagonally in . The resulting polyhedron is the union of the -cell with two opposite protruding edges, each of length .