A property of the planar measure of the lemniscates

Abstract

In this paper, we establish the following conjecture: There exists a constant K such that every lemniscate E(a,c), a Î C^n, c>0, contains a disk B(a, c) with mu(E(a, c)) £ Kmu(B(a, c)), where mu is the planar measure. We prove this conjecture for any family of lemniscates with at the most three foci and for any family of lemniscates where its foci satisfy a suitable condition.