  In the simplest case, consider a $\mathbb{Z}^d$-periodic ($d \geq 3$)
arrangement of balls of radii < 1/2, and select a random direction and point
(outside the balls). According to Dettmann's first conjecture, the probability
that the so determined free flight (until the first hitting of a ball) is
larger than $t >>1$ is $\sim \frac{C}{t}$, where $C$ is explicitly given by the
geometry of the model. In its simplest form, Dettmann's second conjecture is
related to the previous case with tangent balls (of radii 1/2). The conjectures
are established in a more general setup: for $\mathcal{L}$-periodic
configuration of - possibly intersecting - convex bodies with $\mathcal{L}$
being a non-degenerate lattice. These questions are related to P\'olya's
visibility problem (1918), to theories of Bourgain-Golse-Wennberg (1998-) and
of Marklof-Str\"{o}mbergsson (2010-). The results also provide the asymptotic
covariance of the periodic Lorentz process assuming it has a limit in the
super-diffusive scaling, a fact if $d = 2$ and the horizon is infinite.
