As a bonus question in an exam we were asked to find compact metrix spaces $X,Y$ and $Z$ such that $d_{GH}(X,Y)=d_{GH}(X,Z)=d_{GH}(Y,Z)>0$.

The proposed answer is to take $\{0\},\{-1,1\}$ and $\{-1,0,1\}$. And the distances can be easily calculated by trying all appropriate correspondances and calculating distortions.

However I proposed the following three sets ( there are only two radii and they are $R>r$)

set $X$ is the big closed ball of radius $R$.

Set $Y$ is the small closed ball of radius $r$.

Set $Z$ is a closed ball of radius $r$ union two perpendicular line segments of length $2R$ that intersect in the center of the ball.

The conjecture is that all distances are $R-r$. To achieve this distance simply place the figures concentrictly. (I think we may need that $\frac{r}{R}$ is big)

In order to get lower bounds for the distances involving $Y$ one simply uses the bound $d_{GH}(A,B)\geq |\text{Diam}(A)-\text{Diam}(B)|/2$.

But I am stuck calculating the distance between $X$ and $Z$. One approach is to use contradiction and try to use distortions. If we take $x_1$ and $x_2$ diametrally opposite then if $z_1\sim x_1$ and $x_2\sim z_2$ we must have $d(z_1,z_2)>r$ and so at least one of $z_1$ and $z_2$ is outside the small ball, but Im stuck after this.

topologyusing that distance. $\endgroup$ – Igor Rivin Dec 15 '17 at 16:54