Rectifiability of the straight visible boundary and the limits of segment-based Hardy criteria
Weighted Hardy inequalities on a domain $Ω\subset\R^n$ can be deduced from lower Hausdorff-content bounds on suitable visible parts of the boundary, and it is natural to look for visible sets described by straight segments, which are easier to verify than the curve-based visual boundary of Koskela and Lehrbäck. We show that this approach is subject to an intrinsic obstruction. If a boundary point $ξ$ is seen from $x\inΩ$ along a segment satisfying a uniform thickness condition, then $Ω$ contains a truncated cone with vertex $ξ$; consequently the quantitatively straight visible boundary $\partial^{\mathrm{str}}_{x,α}Ω\cap B(x,C_0d_Ω(x))$ is a finite union of Lipschitz graphs, is $(n-1)$-rectifiable, and carries no $λ$-Hausdorff content for any $λ>n-1$. Since content conditions with $λ\le n-1$ already imply the $(p,β)$-Hardy inequality for $β n-1$.