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. © 2007 Elsevier Inc. All rights reserved