A geometric uncertainty principle with an application to Pleijel's estimate

Abstract

Consider partitions of an open, bounded domain in $\mathbb{R}^n$. Then an average element of the partition has either its Fraenkel asymmetry or its deviation from the smallest element in the partition bounded away from 0 by a universal constant. As an application, we give an (unspecified) improvement of Pleijel's estimate on the number of nodal domains of a Laplacian eigenfunction similar to recent work of Bourgain and improve a bound coming from spectral partition problems.