Large Cells and Faces

This chapter is motivated by a conjecture which D.G. Kendall made in the early 1940s. The conjecture concerned a stationary, isotropic Poisson line process in the plane and asked whether the conditional law for the shape of its zero cell, given the area of the zero cell, converges weakly, as the area of the zero cell tends to infinity, to the degenerate law concentrated at the circular shape. An affirmative answer was given by Kovalenko in 1997. Later, very general versions of Kendall’s problem have been studied, and the present chapter gives an account of some of these developments. We give a complete proof for a general version of Kendall’s problem. Other variants will be surveyed. Also on the topic of large faces, we give only a survey, with hints to the proofs in the original literature.