Fekete problem

In mathematics, the Fekete problem is, given a natural number N and a real s &ge; 0, to find the points x1,...,xN on the 2-sphere for which the s-energy, defined by


 * $$ \sum_{1 \leq i < j \leq N} \|x_i - x_j \|^{-s} $$

for s > 0 and by


 * $$ \sum_{1 \leq i < j \leq N} \log \|x_i - x_j \|^{-1} $$

for s = 0, is minimal. For s > 0, such points are called s-Fekete points, and for s = 0, logarithmic Fekete points (see ). More generally, one can consider the same problem on the d-dimensional sphere, or on a Riemannian manifold (in which case ||xi &minus;xj|| is replaced with the Riemannian distance between xi and xj).

The problem originated in the paper by who considered the one-dimensional, s = 0 case, answering a question of Issai Schur.

An algorithmic version of the Fekete problem is number 7 on the list of problems discussed by.