Asymptotic Shapes of large Cells in Random Tessellations

Abstract.

We establish a close relationship between isoperimetric inequalities for convex bodies and asymptotic shapes of large random polytopes, which arise as cells in certain random mosaics in d-dimensional Euclidean space. These mosaics are generated by Poisson hyperplane processes satisfying a few natural assumptions (not necessarily stationarity or isotropy). The size of large cells is measured by a class of general functionals. The main result implies that the asymptotic shapes of large cells are completely determined by the extremal bodies of an inequality of isoperimetric type, which connects the size functional and the expected number of hyperplanes of the generating process hitting a given convex body. We obtain exponential estimates for the conditional probability of large deviations of zero cells from asymptotic or limit shapes, under the condition that the cells have large size.