Peak Set Crossing all the Circles

Let $\Omega\subset\Bbb C^{d}$ be a circular, bounded, strictly convex domain with $C^{2}$ boundary. We construct a peak set $K\subset\partial\Omega$ which intersects all the circles in $\partial\Omega$ with the center at zero. In particular Hausdorff dimension of $K$ is at least $2d-2$.