# Finite approximations to the Kuratowski/Fréchet embedding

Let $$(X,d)$$ be a compact doubling metric space with doubling constant $$C>0$$. Let $$\{\mathbb{X}_n\}_{n=0}^{\infty}$$ be a sequences of finite subsets of $$X$$ with $$\left\{B\left(x_k,\frac1{n}\right)\right\}_{k=0}^{\mathbb{X}_n} \mbox{covers } X \mbox{ and } \#\mathbb{X}_n \mbox{is the \frac1{k}-covering number of X}.$$ Fix some $$x^{\star}\in X$$ and consider the associated sequences of $$1$$-Lipschitz maps $$K_n:\,x\mapsto \left(d(x,x_n)-d(x_n,x^{\star})\right)_{x_n\in \mathbb{X}_n}.$$

In the case where $$(X,d)$$ is a compact Riemannian manifold, then this paper of Katz and Katz (with un unpublished quantitative version found here) shows the Kuratowksi embedding by first showing (in their proof) that the $$K_n$$ approximate the Fréchet embedding $$K_{\infty}:x\mapsto \left(d(x,\cdot)-d(\cdot,x^{\star})\right) \in \ell^{\infty}.$$

What I mean is, if $$(X,d)$$ is compact (and possibly doubling as above) then can we always find sets $$\{\mathbb{X}_n\}_{n=0}^{\infty}$$ such that $$\lim\limits_{n \uparrow \infty}\,\max_{x\in X}\,|K_n(x)-K_{\infty}(x)|=0?$$

I'm assuming so, but I really can't see what the obstruction would be/why I can't find this result in the literature...

For instance, if $$\emptyset\neq X'\subsetneq X$$ and $$X$$ is a compact Riemannian manifold without boundary, then the construction should work by only considering points in $$X'$$. Unless I'm missing something, this should work and I guess it should work for any doubling space by applying Assouad and replacing'' Lipschitz with Hölder?

• It seems that your formulation is wrong --- the functions under max have different domains. It can be made right, and then it is easy to prove. Mar 1, 2022 at 22:54
• @AntonPetrunin It is fixed, the target embedding is the Fréchet one. Mar 2, 2022 at 10:01