Expected distance within a distribution is smaller?

Question

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than the distance between two points draw from different distributions? Assume, $X,Y \sim f_1$, iid and $Z\sim f_2$

$$ E_{X,Y}[dist(X,Y)]\leq^? E_{X,Z}[dist(X,Z)] $$

You can assume any common distance here, Euclidean or $L1$ or ...

Intuitively, for two disjoint distribution, the average distance within the distribution should be smaller than the one across distribution. But I don't know how to prove this simple claim. I came across the Wasserstein metric, where : $$ W_1(f_1,f_2)=\inf E[d(X,Z)^p]^{1/p} $$ it can be shown $W_1(f_1, f_1)=0\leq W_1(f_1,f_2)$, However this is not quite what I want, I don't need the infimum, but my case assumes the distributions are disjoint.

Thanks

Answer 1

Even for uniform distributions on disjoint convex subsets of the plane it is not necessarily true.

For a simple example, let $f_1$ be a uniform distribution on the segment from (0, 0) to (2, 0) and let $f_2$ be the Dirac measure at (1, 1/10). Now $$ E_{X,Y}\|X-Y\|_1 = 2/3 > 3/5 = E_{X,Z}\|X -Z\|_1 $$ in the $\ell^1$ norm. In the Euclidean norm the difference only gets bigger.