mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Fekete problem is, given a natural number ''N'' and a real ''s'' ≥ 0, to find the points ''x''₁,...,''x''_''N'' on the

2-sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...

for which the ''s''-energy, defined by :

\sum_ \, x_i - x_j \, ^

for ''s'' > 0 and by :

\sum_ \log \, x_i - x_j \, ^

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 differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

(in which case , , ''x''_''i'' −''x''_''j'', , is replaced with the Riemannian distance between ''x''_''i'' and ''x''_''j''). The problem originated in the paper by who considered the one-dimensional, ''s'' = 0 case, answering a question of

Issai Schur Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the University of Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at the ...

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

References

* * * *{{Citation , last1=Smale , first1=Stephen , author-link = Stephen Smale , title=Mathematical problems for the next century , doi=10.1007/BF03025291 , mr=1631413 , year=1998 , journal=

The Mathematical Intelligencer ''The Mathematical Intelligencer'' is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes are released quar ...

, issn=0343-6993 , volume=20 , issue=2 , pages=7–15, s2cid=1331144 Mathematical analysis Approximation theory