  We prove some basic theorems concerning lemniscate configurations in an
Euclidean space of dimension $ n \geq 3$. Lemniscates are defined as follows.
Given m points $w_j $ in $\mathbb R^n$, consider the function $F(x)$ which is
the product of the distances $ |x-w_j|$: the singular level sets of the
function $F$ are called lemniscates. We show via complex analysis that the
critical points of $F$ have Hessian of positivity at least $(n-1)$. This
implies that, if $F$ is a Morse function, then $F$ has only local minima and
saddle points with negativity 1. The critical points lie in the convex span of
the points $|w_j| $ (these are absolute minima): but we made also the discovery
that $F$ can also have other local minima, and indeed arbitrarily many. We
discuss several explicit examples. We finally prove in the appendix that all
critical points are isolated.
