The worst visibility walk in a random Delaunay triangulation is $O\left(\sqrt{n}\right)$

Smooth analysis of convex hulls

We establish an upper bound on the smoothed complexity of convex hulls in ${ℝ}^d$ under uniform Euclidean (${\ell }^2$) noise. Specifically, let $\{p_1^{*},p_2^{*},...,p_n^{*}\}$ be an arbitrary set of $n$ points in the unit ball in ${ℝ}^d$ and let $p_i=p_i^{*}+x_i$, where $x_1,x_2,...,x_n$ are chosen independently from the unit ball of radius $\delta$. We show that the expected complexity, measured as the number of faces of all dimensions, of the convex hull of $\{p_1,p_2,...,p_n\}$ is $O\left(n^{2-\frac{4}{d+1}}{\left(1+1/\delta \right)}^{d-1}\right)$; the magnitude $\delta$ of the noise may vary with $n$. For $d=2$ this bound improves to $O\left(n^{\frac{2}{3}}(1+{\delta }^{-\frac{2}{3}})\right)$.