Asymptotic shapes of random polytopes

  • Date: 10/25/2007

Rolf Schneider (University of Freiburg)


University of Alberta


We consider random polytopes, generated as intersections of closed
halfspaces (containing 0) bounded by the hyperplanes of a Poisson
process of hyperplanes (satisfying only some homogeneity property under
dilatations). The central question (a very general version of D.G.
Kendall's problem) asks for the asymptotic shape of the random polytope
under the condition that it is large (measured in various ways). The
answer depend on the extremal bodies of inequalities of isoperimetric
type for certain functionals of convex bodies, and stability results
for these lead to estimates for probabilities of large deviations from
asymptotic shapes. (Joint work with Daniel Hug and partially with
Matthias Reitzner)

Other Information: 

PIMS Distinguished Lecture 2007