Peak Set Crossing all the Circles

Let

be a circular, bounded, strictly convex domain with

C^{2} C 2

boundary. We construct a peak set

K\subset\partial\Omega K ⊂ ∂ Ω

which intersects all the circles in

\partial\Omega ∂ Ω

with the center at zero. In particular Hausdorff dimension of

K K

is at least

2d-2 2 d − 2

.