2015/06/05 by Ingrid Bauer, Bauer, Ingrid, Fabrizio Catanese +4 · 1 citation
Mathematics · #14P25 #32S50 #32U05 #37EXX #51F99 #68U05 #70F99 #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Metric Geometry (math.MG) #Point processes and geometric inequalities #math.AG #math.CV #math.MG #msc:14P25 #msc:32S50 #msc:32U05 #msc:37EXX #msc:51F99 #msc:68U05 #msc:70F99
paper · pdf · doi:10.48550/arxiv.1506.01919
31 pages,3 figures, to appear in the Annali della Scuola Normale Superiore. In the new version we show that all critical points are isolated
openalex publication_date 2015/06/05 · arxiv created 2017/05/18 · arxiv updated 2017/05/22 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We prove some basic theorems concerning lemniscate configurations in an Euclidean space of dimension n ≥ 3. Lemniscates are defined as follows. Given m points wj in \mathbb Rn, consider the function F(x) which is the product of the distances |x-wj|: 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 |wj| (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.