  We consider the convex hull of the perturbed point process comprised of $n$
i.i.d. points, each distributed as the sum of a uniform point on the unit
sphere $\S^{d-1}$ and a uniform point in the $d$-dimensional ball centered at
the origin and of radius $n^{\alpha}, \alpha \in (-\infty, \infty)$. This
model, inspired by the smoothed complexity analysis introduced in computational
geometry \cite{DGGT,ST}, is a perturbation of the classical random polytope. We
show that the perturbed point process, after rescaling, converges in the
scaling limit to one of five Poisson point processes according to whether
$\alpha$ belongs to one of five regimes. The intensity measure of the limit
Poisson point process undergoes a transition at the values $\alpha = \frac{-2}
{d -1}$ and $\alpha = \frac{2} {d + 1}$ and it gives rise to four rescalings
for the $k$-face functional on perturbed data. These rescalings are used to
establish explicit expectation asymptotics for the number of $k$-dimensional
faces of the convex hull of either perturbed binomial or Poisson data. In the
case of Poisson input, we establish explicit variance asymptotics and a central
limit theorem for the number of $k$-dimensional faces. Finally it is shown that
the rescaled boundary of the convex hull of the perturbed point process
converges to the boundary of a parabolic hull process.
