Definition of pointed Gromov-Hausdorff convergence for metric spaces

Question

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 A A 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

1. $f(p_n)=p$
2. $\text{dis} f<\varepsilon$ (i.e. $\sup_{B(p_n,r)}|d_n(x_1,x_2)-d(f(x_1),f(x_2))|<\varepsilon$)
3. 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 $X\sqcup Y$ which extend the metrics on $X$ and $Y$ and $d_H$ denotes the Hausdorff distance of $X$ and $Y$ as subsets of $X\sqcup Y$ and gives the following definition

Definition B A 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 $r_n\rightarrow r$ 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 C Let $(X, d_X , p)$ and $(X_n, d_{X_n} , p_n)$, $n\in\mathbb{N}$, 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 $(\overline{B}_{X_n}(0,1),0)\not\to (\overline{B}_X(0,1),0)$ and $(X_n,0)\not\to (X,0)$ according to definition C.

Also, these notes (which adopt Petersen's definition B) claim that if $\{(X_n,d_n,p_n)\}$ ($n\in\mathbb{N}$) 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 $r_n\to r$ 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?