A note on the Erdös minimal area problem

Abstract

We answer a question of Erdős, Herzog, and Piranian \cite[Problem $4$, page $135$]{EHP} on the minimal area of polynomial lemniscates when all the zeros of the polynomial are constrained to lie on a compact set $K$ whose logarithmic capacity is greater than $1$. We show that the minimal area decays exponentially fast and prove that the Fekete polynomials are asymptotic minimizers for the area problem.

14 pages, corrected some typos. Some new results added. Accepted for publication in the Proceedings of the AMS,