On the maximum of the minimum of a sum (Q580412)

Let B denote the closure of a bounded open set of points in \(E^ n\) with Jordan content \(| B| >0\) and let \(c>0\) be constant. Typical of the expressions considered is \[ M(N,c)=\max_{\{x_ j\}}\min_{x\in B}\sum^{N}_{j=1}| x-x_ j|^{-c},\quad x_ j\in E^ n\quad. \] Together with its analogs and extensions, the problem for \(c<n\) has a long history, associated with the names of Fekete, Leja, Pólya, Szegö, Frostman and Carleson, to mention just a few. It involves the notions of generalized capacity, transfinite diameter, and equilibrium potential. Here we consider the case \(c\geq n\) and its extensions, for which the prior history seems less comprehensive. Illustrative of the results obtained are the three equations \[ \lim_{N\to \infty}\frac{M(N,n)}{N \log N}=\frac{\omega (n)}{| B|},\quad \lim_{N\to \infty}\frac{M(N,c)}{N^{c/n}}=\frac{L(c,n)}{| B|^{c/n}}, \] \[ \lim_{c\to \infty}L(n,c)^{1/c}=\lim_{N\to \infty}\frac{(| B| /N)^{1/n}}{\rho (N)}. \] In the first \(c=n\) and \(\omega\) (n) is the volume of the unit ball. In the second \(c>n\) and existence of the limit is asserted, \(0<L(n,c)<\infty\). In the third, \(\rho\) (N) is the smallest value such that N spheres of radius \(\rho\) (N) can cover B. The results would be unchanged if we required \(x_ j\in B\) instead of \(x_ j\in E^ n\) in the definition of M(N,c).