Whereas the definition of Gromov-Hausdorff convergence for compact metric spaces seems to be standard, difference sources seem to give slightly different definitions of pointed Gromov-Hausdorff convergence for (noncompact) pointed metric spaces.

For example, *A course in metric geometry* (D. Burago, Yu. Burago, and S. Ivanov) at page 272 gives this definition

Definition AA sequence ${(X_n,d_n,p_n)}$ of pointed metric spaces converges to the pointed metric space ${(X,d,p)}$ if for all $varepsilon,r>0$ there exists a natural number $n_0$ such that for every $n>n_0$ there is a map $f:B(p_n,r)rightarrow X$ such that

- $f(p_n)=p$
- $text{dis} f<varepsilon$ (i.e. $sup_{B(p_n,r)}|d_n(x_1,x_2)-d(f(x_1),f(x_2))|<varepsilon$)
- the $varepsilon$-neighbourhood of $f(B(p_n,r))$ contains the ball $B(p,r-varepsilon)$

On the other hand, Petersen’s *Riemannian Geometry* (3rd edition) in Chapter 11.1.2 (page 401) restricts to **proper** metric spaces, introduces a pointed Gromov-Hausdorff distance as follows:

$$d_{text{GH}}left((X,p),(Y,q)right)=inf{d_H(X,Y)+d(p,q)}$$

where the inf is over all metrics $d$ on the disjoint union $Xsqcup Y$ which extend the metrics on $X$ and $Y$ and $d_H$ denotes the Hausdorff distance of $X$ and $Y$ as subsets of $Xsqcup Y$ and gives the following definition

Definition BA sequence ${(X_n,d_n,p_n)}$ of pointed metric spaces converges to the pointed metric space ${(X,d,p)}$ if for all $r>0$ there exists a sequence $r_nrightarrow r$ such that

$$d_{text{GH}}left((overline{B}(p_n,r_n),p_n),(overline{B}(p,r),p)right)rightarrow 0$$

Finally, Gromov’s own book *Metric Structures for Riemannian and Non-Riemannian Spaces* (Def. 3.1.4, at page 85) uses essentially the same definition as Petersen’s, but restricts to (complete) locally compact length metric spaces.

And these are not all the definitions I have found in the literature.

For example, this paper by Dorothea Jansen states (Definition 2.1)

Definition CLet $(X, d_X , p)$ and $(X_n, d_{X_n} , p_n)$, $ninmathbb{N}$, be pointed proper

metric spaces. If $$d_{text{GH}}left((overline{B}(p_n,r),p_n),(overline{B}(p,r),p)right)rightarrow 0$$ for all $r>0$ where the balls are equipped with the restricted metric, then $(X_n, p_n)$ converges to $(X, p)$ in the pointed Gromov-Hausdorff sense.

Not all of them are equivalent. For example, say $X_n$ is the space consisting of the two points ${0,1+frac{1}{n}}$ and $X={0,1}$ (both with the metric inherited from $mathbb{R}$).

Here all $X_n$ (and $X$) are proper, but they are not length spaces. It is easy to see that $(X_n,0)rightarrow (X,0)$ according to definitions A and B. However for all $n$, $overline{B}_{X_n}(0,1)={0}$ and $overline{B}_X(0,1)={0,1}$, so $(overline{B}_{X_n}(0,1),0)notto (overline{B}_X(0,1),0)$ and $(X_n,0)notto (X,0)$ according to definition C.

Also, these notes (which adopt Petersen’s definition B) claim that if ${(X_n,d_n,p_n)}$ ($ninmathbb{N}$) and ${(X,d,p)}$ are **compact** pointed spaces such that $d_{text{GH}}left((X_n,p_n),(X,p)right)to 0$, then $(X_n,p_n)to (X,p)$ in the sense of definition B, i.e. for each $r>0$ there exist $r_nto r$ such that $$d_{text{GH}}left((overline{B}(p_n,r_n),p_n),(overline{B}(p,r),p)right)rightarrow 0$$

without ever assuming the spaces are length spaces.

My questions are:

- When are these definitions equivalent?
- What are the advantages of restricting to proper spaces? What are the advantages of restricting to length spaces?
- Is there a reference which deals with these issues?