A new estimate of the transfinite diameter of Bernstein sets

Abstract

Let $K \subset \mathbb{C}^n$ be a compact set satisfying the following Bernstein inequality: for any $m \in \{ 1,..., n\}$ and for any $n$-variate polynomial $P$ of degree $\mbox{deg}(P)$ we have \begin{align*} \max_{z\in K}\left|\frac{\partial P}{\partial z_m}(z)\right| \le M\ \mbox{deg}(P) \max_{z\in K}|P(z)| \ \mbox{ for } z = (z_1, \dots, z_n). \end{align*} for some constant $M= M(K)>0$ depending only on $K$. We show that the transfinite diameter of $K$, denoted $\delta(K)$, verifies the following lower estimate \begin{align*} \delta(K) \ge \frac{1}{n M}, \end{align*} which is optimal in the one-dimensional case. In addition, we show that if $K$ is a Cartesian product of compact planar sets then \begin{align*} \delta(K) \ge \frac{1}{M}. \end{align*}