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

TL;DR

This paper explores the geometric properties of lemniscates formed by logarithmic potentials in higher-dimensional Euclidean spaces, revealing critical point behavior and the possibility of multiple local minima.

Contribution

It extends the understanding of lemniscate configurations to higher dimensions and proves new theorems about their critical points and minima.

Findings

Critical points have Hessian positivity at least (n-1).

F has only local minima and saddle points with negativity 1 if Morse.

F can have arbitrarily many local minima.

Abstract

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.