Here is some (non-definitive) experimental data.
The figure below shows $10$ random points in each of $X,Y,Z$, 
and the minimum perimeter $\triangle$:
<hr />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
[![XYZ][1]][1]
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<sup>
$|X|=|Y|=|Z|=10$. Minimum (red-green-blue) perimeter $\triangle$ drawn.
</sup>

Now here I let $n$ vary, with $|X|=|Y|=|Z|=n$,
and average the results over $k$ trials:
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
[![Plot_30][2]][2]
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<sup>
The average min perimeter over $50$ random trials.
Fit: $0.03\, + \frac{2.5}{n}$.
</sup>
<hr />
The data seems to fit $O(\frac{1}{n})$, but I would not want to claim 
that $n=30$
settles the asymptotics.


  [1]: https://i.sstatic.net/c7zmQ.jpg
  [2]: https://i.sstatic.net/EWLJM.jpg