Here is some (non-definitive) experimental data. The figure below shows $n{=}10$ random points in each of $X,Y,Z$, and the minimum perimeter $\triangle$:

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          [![XYZ][1]][1]
          ^{$|X|=|Y|=|Z|=10$. Minimum (red-green-blue) perimeter $\triangle$ drawn.}

Now here I let $n$ vary, with $|X|=|Y|=|Z|=n$, and average the results over $k$ trials:

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          [![n50][2]][2]
          ^{The average min perimeter over $k{=}50$ random trials. Fit: $0.03\, + \frac{2.53}{n}$.}

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The data seems to fit $O(\frac{1}{n})$, but I would not want to claim that $n{=}50$ settles the asymptotics.

Here is some experimental data. The figure below shows $n{=}10$ random points in each of $X,Y,Z$, and the minimum perimeter $\triangle$:

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          [![XYZ][1]][1]
          ^{$|X|=|Y|=|Z|=10$. Minimum (red-green-blue) perimeter $\triangle$ drawn.}

Now here I let $n$ vary, with $|X|=|Y|=|Z|=n$, and average the results over $k$ trials:

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          [![Fit][2]][2]
          ^{The average min perimeter over $k{=}50$ random trials. Fit: $1.54 \,/\, n^{3/4}$.}

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The data fits Ori Gurel-Gurevich's calculation reasonably well.

Here is some (non-definitive) experimental data. The figure below shows $10$ random points in each of $X,Y,Z$, and the minimum perimeter $\triangle$:

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          [![XYZ][1]][1]
          ^{$|X|=|Y|=|Z|=10$. Minimum (red-green-blue) perimeter $\triangle$ drawn.}

Now here I let $n$ vary, with $|X|=|Y|=|Z|=n$, and average the results over $k$ trials:

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          [![Plot_30][2]][2]
          ^{The average min perimeter over $50$ random trials. Fit: $0.03\, + \frac{2.5}{n}$.}

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The data seems to fit $O(\frac{1}{n})$, but I would not want to claim that $n=30$ settles the asymptotics.

Here is some (non-definitive) experimental data. The figure below shows $n{=}10$ random points in each of $X,Y,Z$, and the minimum perimeter $\triangle$:

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          [![XYZ][1]][1]
          ^{$|X|=|Y|=|Z|=10$. Minimum (red-green-blue) perimeter $\triangle$ drawn.}

Now here I let $n$ vary, with $|X|=|Y|=|Z|=n$, and average the results over $k$ trials:

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          [![n50][2]][2]
          ^{The average min perimeter over $k{=}50$ random trials. Fit: $0.03\, + \frac{2.53}{n}$.}

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The data seems to fit $O(\frac{1}{n})$, but I would not want to claim that $n{=}50$ settles the asymptotics.