  We prove a randomized version of the generalized Urysohn inequality relating
mean-width to the other intrinsic volumes. To do this, we introduce a
stochastic approximation procedure that sees each convex body K as the limit of
intersections of Euclidean balls of large radii and centered at randomly chosen
points. The proof depends on a new isoperimetric inequality for the intrinsic
volumes of such intersections. If the centers are i.i.d. and sampled according
to a bounded continuous distribution, then the extremizing measure is uniform
on a Euclidean ball. If one additionally assumes that the centers have i.i.d.
coordinates, then the uniform measure on a cube is the extremizer. We also
discuss connections to a randomized version of the extended isoperimetric
inequality and symmetrization techniques.
