Longest visible rays in Boolean models with general convex grains

We consider a stationary Boolean model in R d whose grains are random compact convex bodies with non-empty interior. Conditionally on the origin being uncovered, we study the logarithmic tail asymptotics of the length L o of the longest visible ray starting at the origin. For a fixed direction u ∈ S d − 1 , the visible range has an exponential tail with rate equal to the intensity times the mean ( d − 1 ) -dimensional volume of the projection of the typical grain onto u ⊥ . Optimizing over all directions therefore leads to an anisotropic variational principle. We show that the decay rate for L o is determined by the minimum mean brightness of the typical grain.