Random ball-polyhedra and inequalities for intrinsic volumes Academic Article uri icon

abstract

  • © 2016, Springer-Verlag Wien. 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.

author list (cited authors)

  • Paouris, G., & Pivovarov, P.

citation count

  • 7

publication date

  • March 2017