All-set-homogeneous spaces

Question

This is a follow-up to the question of Joseph O'Rourke Which metric spaces have this superposition property?

A metric space $X$ will be called all-set-homogeneous if for any subset $A\subset X$ any distance-preserving map $A\to X$ can be extended to an isometry $X\to X$.

Classical examples include Euclidean spaces, Lobachevsky spaces, and spheres (all with canonical metrics and up to rescaling).

Is it true that complete all-set-homogeneous length-metric spaces include only the classical examples plus the so-called universal $\mathbb{R}$-trees of finite valence?

We have a partial answer, which requires the following definition.

Given a metric space $X$, consider all pseudometrics induced on $n$ points $x_1,\dots, x_n\in X$. Such a metric is completely described by $n{\cdot}(n-1)/2$ real numbers $|x_i-x_j|_X$, so it can be encoded by a point in $\mathbb{R}^{n{\cdot}(n-1)/2}$. The set $F_n(M)\subset \mathbb{R}^{n{\cdot}(n-1)/2}$ of all these points is called the $n^\text{th}$ fingerprint of $X$.

We know that all-set-homogeneous length-metric spaces with closed fingerprints include only the basic examples. (We do not assume local compactness.) The proof is very short, you may check it here, but likely it is not new.

Is it new?

Remarks

• Locally compact all-set-homogeneous length spaces include only basic examples; in fact, in this case, 3-point-homogeneity is sufficient. It follows from the result of J. Tits ["Sur certaines classes d'espaces homogènes de groupes de Lie" (1955)];

• It seems that condition that the space is length can not be seriously connected to all-set-homogeneity. So we expect that there are counterexamples to the first question.

• It was shown by Birkhoff that all-set-homogeneous geodesic space with local uniqueness of geodesics has to be classical [See his Metric foundations of geometry. I].

Answer 1

This is more of an observation and a long comment than an answer.

The observation is that a fairly standard application of the Erdős–Rado theorem implies that in any metric space with more than $2^{2^{\aleph_0}}$ many elements, there is an infinite sequence $(a_i)_{i<\omega}$ of distinct elements satisfying $d(a_i,a_j) = d(a_0,a_1)$ for any $i<j<\omega$. By the argument in your note this is enough to imply that any all-set-homogeneous metric space (with no assumption of completeness or length-ness) has cardinality at most $2^{2^{\aleph_0}}$.

The comment regards connections between your question and concepts that have been studied in model theory, particularly in the model theory of metric structures.

In your note, you mention a variation of this question in which you only require homogeneity over finite sets. In discrete structures, this kind of homogeneity is studied in model theory and is referred to as ultrahomogeneity (as opposed to ordinary homogeneity, which requires a stronger relationship between the two finite sets of elements to get an automorphism). The relevant concept here is that of a Fraïssé class, which is a class of finite structures satisfying the properties of the class of finite substructures of an ultrahomogeneous structure. Fraïssé's theorem is the statement that it is possible to construct the structure itself from the class in a particular way. In particular, countable ultrahomogeneous structures (in a finite relational language) are precisely characterized as the Fraïssé limits of Fraïssé classes (although this doesn't necessarily help that much with actually characterizing such structures precisely).

In my subfield of model theory, known as continuous logic, we apply model-theoretic techniques to structures with underlying metrics, such as metric spaces, Banach spaces, $\mathbb{R}$-trees, and so on. Fraïssé limits have been studied in the context of continuous logic by Ben Yaacov, although frankly the necessary formalism is quite technical when compared to the discrete version. As is often the case in continuous logic, some of the defining properties of Fraïssé classes need to be modified to only hold in some approximate sense. Here the natural notion is that of approximate ultrahomogeneity. In particular, a metric space $X$ is approximately ultrahomogeneous if for any finite $A \subseteq X$, any distance-preserving map $f : A \to X$, and any $\varepsilon > 0$, there is an isometry $g : X \to X$ such that $d(f(a),g(a)) < \varepsilon$ for every $a \in A$.

So, using Ben Yaacov's machinery, I am able to give a technical and possibly useless answer to a modification of a question you did not ask here, which is that the separable approximately ultrahomogeneous metric spaces are precisely the limits of Fraïssé classes (in the sense of Ben Yaacov) of finite metric spaces.

Unfortunately, the all-set-homogeneity property doesn't have analogs that have been studied by model theorists to my knowledge. The issue is that, as we already saw, properties of this sort tend to limit the size of the structure involved, and this makes it a little awkward to use model-theoretic tools.

Finally, I can comment on the axiomatizability of the class of length spaces in continuous logic. A fairly well-known fact is that complete length spaces can be characterized as those complete metric spaces that admit approximate midpoints (i.e., for any $a,b \in X$ and any $\varepsilon > 0$, there is a $c \in X$ such that $d(a,c)$ and $d(c,b)$ are both within $\varepsilon$ of $\frac{1}{2}d(a,b)$. This condition can be easily expressed in continuous logic by the closed condition $\sup_{xy}\inf_{z} \max(|d(x,z)-\frac{1}{2}d(x,y)|,|d(z,y)-\frac{1}{2}d(x,y)|) = 0$, so for any $r > 0$, the class of (complete) length spaces of diameter at most $r$ is elementary (in the sense of continuous logic, which by default only deals with complete metric structures). Dealing with unbounded metric spaces, while doable in continuous logic, is a bit trickier. It can be accomplished using Ben Yaacov's extension of continuous logic to unbounded metric structures. In this situation too, it is possible to axiomatize the class of length spaces, but it's a little bit more complicated to state how precisely.