I will describe the question first in 2D, but my interest is in $\mathbb{R}^3$. An electron $x$ will shoot from the origin along an initial vector $v$. You know the speed $|v|$ but not the direction. Your task is to arrange a minimum number of point-charge electrons at fixed locations on an origin-centered unit circle so that, no matter in which direction $v$ is pointing, $x$ cannot escape the disk. All the charges are equal, and repel each other via inverse-square Coulomb force.

Below there are four charges on the circle. The left electron escapes, the right, which is shot along the same direction but with a lesser speed, does not. (Of course aiming $v$ along a diagonal is the best escape strategy in this configuration.)

Now here is my question. Consider the same problem in $\mathbb{R}^3$.
Given $|v|$, arrange fixed electrons on a sphere to cage-in the electron $x$
starting from the origin with velocity $v$ in any direction.
It seems natural to think that an optimal arrangement is a type
of disk-packing on a sphere,
for example, the solutions to the
*Thompson problem*
or the
*Tammes problem*.
(Here I am mining an earlier MO question.)

^{ (Image from Paul Sutcliffe.) }

Q. Is an optimal electron cage for a given speed $|v|$ a configuration that minimizes electrostatic potential? Or maximizes the minimum distance between any pair of electrons, i.e., an optimal disk packing on the sphere?

**by Robert Israel: No such cage is possible in $\mathbb{R}^3$!**

*Answered*($\mathbb{R}^2$ is rather different and not a reliable guide to $\mathbb{R}^3$.)