On the Spacing of Fekete Points for a Sphere, Ball or Simplex

Suppose that $K\subset\RR^d$ is either the unit ball, the unit sphere or the standard simplex. We show that there are constants $c_1,c_2>0$ such that for a set of Fekete points (maximizing the Vandermonde determinant) of degree $n,$ $F_n\subset K,$ $${c_1\over n}\le \min_{b\in F_n\atop b\neq a} \hbox{dist}(a,b)\le {c_2\over n}$$ for all $a\in F_n.$ Here $\hbox{dist}(a,b)$ is a natural distance on $K$ that will be described in the text