Analytic capacity and projections

In this paper we study the connection between the analytic capacity of a set and the size of its orthogonal projections. More precisely, we prove that if $E\subset \mathbb{C}$ is compact and $\mu$ is a Borel measure supported on $E$, then the analytic capacity of $E$ satisfies $$\gamma (E)\ge c\frac{\mu (E{)}^2}{{\int }_I\mathrm{\parallel }{P}_{\theta }\mu {\mathrm{\parallel }}_2^{2}d\theta },$$ where $c$ is some positive constant, $I\subset [0,\pi )$ is an arbitrary interval, and $P_{\theta }\mu$ is the image measure of $\mu$ by $P_{\theta }$, the orthogonal projection onto the line $\{r{e}^{i\theta }:r\in \mathbb{R}\}$. This result is related to an old conjecture of Vitushkin about the relationship between the Favard length and analytic capacity. We also prove a generalization of the above inequality to higher dimensions which involves related capacities associated with signed Riesz kernels.