  We show that for any compact set $E\subset\mathbb{R}^d$ the visible part of
$E$ has Hausdorff dimension at most $d-1/6$ for almost every direction. This
improves recent estimates of Orponen and Matheus. If $E$ is $s$-Ahlfors
regular, where $s>d-1$, we prove a much better estimate. In that case for
almost every direction the Hausdorff dimension of the visible part is at most
$s - \alpha(s-d+1),$ where $\alpha>0.183$ is absolute. The estimate is new even
for self-similar sets satisfying the open set condition. Along the way, we
prove a refinement of the Marstrand's slicing theorem for Ahlfors regular sets.
