  We investigate an extension of an equilibrium-type result, conjectured by
Ambrus, Ball and Erd\'elyi, and proved recently by Hardin, Kendall and Saff.
These results were formulated on the torus, hence we also work on the torus,
but one of the main motivations for our extension comes from an analogous setup
on the unit interval, investigated earlier by Fenton.
  Basically, the problem is a minimax one, i.e. to minimize the maximum of a
function $F$, defined as the sum of arbitrary translates of certain fixed
"kernel functions", minimization understood with respect to the translates. If
these kernels are assumed to be concave, having certain singularities or cusps
at zero, then translates by $y_j$ will have singularities at $y_j$ (while in
between these nodes the sum function still behaves realtively regularly). So
one can consider the maxima $m_i$ on each subintervals between the nodes $y_j$,
and look for the minimization of $\max F = \max_i m_i$.
  Here also a dual question of maximization of $\min_i m_i$ arises. This type
of minimax problems were treated under some additional assumptions on the
kernels. Also the problem is normalized so that $y_0=0$.
  In particular, Hardin, Kendall and Saff assumed that we have one single
kernel $K$ on the torus or circle, and $F=\sum_{j=0}^n K(\cdot-y_j)= K +
\sum_{j=1}^n K(\cdot-y_j)$. Fenton considered situations on the interval with
two fixed kernels $J$ and $K$, also satisfying additional assumptions, and $F=
J + \sum_{j=1}^n K(\cdot-y_j)$. Here we consider the situation (on the circle)
when \emph{all the kernel functions can be different}, and $F=\sum_{j=0}^n
K_j(\cdot- y_j) = K_0 + \sum_{j=1}^n K_j(\cdot-y_j)$. Also an emphasis is put
on relaxing all other technical assumptions and give alternative, rather
minimal variants of the set of conditions on the kernel.

Classical Analysis and ODEs