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Let $(X_i,p_i)$, $(X,p)$ be pointed connected proper metric spaces (i.e. the closures of balls are compact). Are the following two statements equivalent?

1. $\forall r > 0: \bar{B}_r(p_i) \stackrel{GH}{\to} \bar{B}_r(p)$.
2. $\forall r > 0: (\bar{B}_r(p_i),p_i) \stackrel{GH}{\to} (\bar{B}_r(p),p)$.

In 2, the pointed Gromov-Hausdorff distance is defined as usual but with respect to the pointed Hausdorff distance $d_H((A,a),(B,b)) := d_H(A,B) + d(a,b)$.

Obviously, 2 implies 1, so the question is whether or not the other implication holds.

Let $(X_i,p_i)$, $(X,p)$ be pointed connected proper metric spaces (i.e. the closures of balls are compact). Are the following two statements equivalent?

1. $\forall r > 0: \bar{B}_r(p_i) \stackrel{GH}{\to} \bar{B}_r(p)$.
2. $\forall r > 0: (\bar{B}_r(p_i),p_i) \stackrel{GH}{\to} (\bar{B}_r(p),p)$.

In 2, the pointed Gromov-Hausdorff distance is defined as usual but with respect to the pointed Hausdorff distance $d_H((A,a),(B,b)) := d_H(A,B) + d(a,b)$.

Obviously, 2 implies 1, so the question is whether or not/under which conditions the other implication holds.

Let $(X_i,p_i)$, $(X,p)$ be pointed proper metric spaces (i.e. the closures of balls are compact). Are the following two statements equivalent?

1. $\forall r > 0: \bar{B}_r(p_i) \stackrel{GH}{\to} \bar{B}_r(p)$.
2. $\forall r > 0: (\bar{B}_r(p_i),p_i) \stackrel{GH}{\to} (\bar{B}_r(p),p)$.

In 2, the pointed Gromov-Hausdorff distance is defined as usual but with respect to the pointed Hausdorff distance $d_H((A,a),(B,b)) := d_H(A,B) + d(a,b)$.

Obviously, 2 implies 1, so the question is whether or not the other implication holds.

Let $(X_i,p_i)$, $(X,p)$ be pointed connected proper metric spaces (i.e. the closures of balls are compact). Are the following two statements equivalent?

1. $\forall r > 0: \bar{B}_r(p_i) \stackrel{GH}{\to} \bar{B}_r(p)$.
2. $\forall r > 0: (\bar{B}_r(p_i),p_i) \stackrel{GH}{\to} (\bar{B}_r(p),p)$.

In 2, the pointed Gromov-Hausdorff distance is defined as usual but with respect to the pointed Hausdorff distance $d_H((A,a),(B,b)) := d_H(A,B) + d(a,b)$.

Obviously, 2 implies 1, so the question is whether or not the other implication holds.