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We prove some basic theorems concerning lemniscate configurations in anEuclidean 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 isthe product of the distances $ |x-w_j|$: the singular level sets of thefunction $F$ are called lemniscates. We show via complex analysis that thecritical points of $F$ have Hessian of positivity at least $(n-1)$. Thisimplies that, if $F$ is a Morse function, then $F$ has only local minima andsaddle points with negativity 1. The critical points lie in the convex span ofthe points $|w_j| $ (these are absolute minima): but we made also the discoverythat $F$ can also have other local minima, and indeed arbitrarily many. Wediscuss several explicit examples. We finally prove in the appendix that allcritical points are isolated.

Higher dimensional Lemniscates: the geometry of $r$ particles in $n$-space with logarithmic potentials