A property of the planar measure of the lemniscates

In this paper, we establish the following conjecture: There exists a constant K such that every lemniscate E(α,c), α∈Cn, c>0, contains a disk B(α,c) with μ(E(α,c))⩽Kμ(B(α,c)), where μ 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.