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