In connection with a conjecture stated by D. G. Kendall in the forties, we describe the asymptotic behavior of the distribution function of the area of the planar Crofton cell. We deduce from this (in support of his conjecture) that expressed in terms of eigenvalues, the large Crofton cells are nearly circular. We obtain also the asymptotic behavior of the Laplace transform of the law of the perimeter of the convex hull of planar Brownian motion run until time 1. This last result implies that the small convex hulls of Brownian motion are nearly circular.
"Sur une conjecture de D. G. Kendall concernant la cellule de Crofton du plan et sur sa contrepartie brownienne." Ann. Probab. 26 (4) 1727 - 1750, October 1998. https://doi.org/10.1214/aop/1022855880