What is $s$ in s-energy (eg. Riesz s-energy)

Question

I'm trying to understand fekete problems. There is a variable $s$ and a related concept of 's-energy' [1] [2] [3] [4] that comes up repeatedly when borrowing the concept of potential energy to find well separated point sets on a sphere. The problem is, though I can find some references to 's-energy', I can't find a definition of $s$ and don't know how to determine its value.

Answer 1

The choice of $s$ is up to you; different choices of $s$ will produce different energy-minimizing configurations of points. The larger $s$ is, the more we penalize the smallest distances between points (without paying that much attention to the distances that are not smallest). In the limit $s\to\infty$ the energy minimization problem becomes the maxmin problem $$\min_{i<j} |x_i-x_j|\to\max $$ If you take inspiration from electrostatics, thinking of points as unit charges that repel one another, then you should use $s=N-2$ where $N$ is the dimension of the ambient space. This is known as Newtonian or Couloumbian energy. The value $s=N-2$ is the one most often used in potential theory.

On the other hand, for computation of derivatives it may help to have $s$ that is an even integer, because then $|x_i-x_j|^{-s}$ is a rational function: square root cancels out.