Lebesgue spine

Counterexample

In $\R^3$, write points as $x = (x_1, x_2, x_3)$ and consider the closed thin spike $$K = \set{x \in \R^3 : 0 < x_1 \le 1, \; x_2^2 + x_3^2 \le e^{-2/x_1}} \cup \set{0}.$$ Its cross-section at distance $x_1$ from the origin is a disc of radius $e^{-1/x_1}$. Let $\Omega = B \setminus K$, where $B$ is the open ball of radius $2$ centred at $0$. The origin $0$ is a boundary point of $\Omega$, the tip of the spike, and it is not a regular boundary point. Check with Wiener's criterion, using the part $E_k$ of the complement in the shell $2^{-k-1} \le \abs{x} \le 2^{-k}$. For large $k$, $E_k$ lies in a thin cylinder of length about $2^{-k-1}$ and radius at most $e^{-2^k}$. A cylinder of length $\ell$ and radius $r \le \ell/2$ has Newtonian capacity at most a constant times $\ell / \log(\ell / r)$. So $\operatorname{cap}(E_k)$ is at most a constant times $2^{-k} / 2^k = 4^{-k}$, and $\sum_k 2^k \operatorname{cap}(E_k)$ converges. Hence $0$ is irregular.