Assume you have a set of 3D vectors $V$ such that the average angle between any 2 vectors in the set is $\theta$.
You want an operation $f(v)$ that will be applied to each vector such that afterwards the mean inner angle is different.
The intuitive way to describe this is, imagine you have a bunch of lazers that you arange such that they point in different directions. Imagine you change all lazers such that they point all point towards roughly the same direction, with some variability. This looks like a chaotic "cone" of red lines starting at the same point and moving outwards.
Or in a diagram:
The transformation $f$ should such that it uses a single scalar to determine whether the transformation will spread or concentrate the rays (e.g 0.9 concentrates them, 1.1 spreads them).
So far the only approach I have is rotate each pair of vectors along their normal to either increase or decrease their inner angle. But I wonder if there is a simpler/better way to do this.
I suggest first calculating the average direction vector $v = \frac1{\|u\|}u$ where $u = \sum_{k=1}^n v_k$, then setting $$f(\lambda, w) = \left(\cos \frac\pi 2\lambda\right)\|w\|v + \left(\sin \frac \pi 2\lambda\right)w$$
Then for any $w, f(0, w) = v, f(1, w) = w, f(2,w) = -v$.
A more easily calculated transformation would be $$f(\lambda, w) = \lambda w + (1 - \lambda) v$$
In the first transformation, the angle changes linearly with $\lambda$, so as seen from the origin, $f(\lambda, w)$ would move at constant rate towards $v$ as $\lambda$ changes. In the second transformation, $f(\lambda, w)$ would move at a similar rate near $\lambda = 0$, but slow down the farther $\lambda$ is from $0$.
Which approach would be best for you depends on what you want to do with it.