Let $L$ and $R$ be two bags of positive vectors such that all vectors have length $k$. Define the distance $d_{avg}$ between the bags as the average pairwise angle between the vectors.
Is is possible to find a vector $l$ (using the vectors from $L$) and a vector $r$ (using the vectors from $R$), which have the same angle (distance) between them? Or alternatively, two constant sized structures from which the average angle can be somehow computed (perhaps using $|L|$ and $|R|$).
Further, if $L=L_l \cup L_r$, is it possible to compute $l$ given only $l_l$ and $l_r$? If needed, $|L_l|$ and $|L_r|$ are available.
Note that taking the average of the vectors in $L$ and $R$ does not work. For instance $L=[[0,1], [0,1], [1,1]]$ and $R=[[1,0]]$. The average of $L$ is $[1/3, 1]$ and the one of $R$ is $[1,0]$. Now, while $d_{avg}(L,R) = \frac{5\pi}{12} = 1.3090 $, the angle between the averages is $1.2490$.
A generalization which allows vectors with negative components might be interesting as well, but it seems like this has many issues of its own.