  We study the incoming boundary value problem for the stationary linearized
Boltzmann equation in bounded convex domains. The geometry of the domain has a
dramatic effect on the space of solutions. We prove the existence of solutions
in $W^{1,p}$ spaces for $1 \leq p<2$ for small domains. In contrast, if we
further assume the positivity of the Gaussian curvature on the boundary, we
prove the existence of solutions in $W^{1, p}$ spaces for $1 \leq p < 3$
provided that the diameter of the domain is small enough. In both cases, we
provide counterexamples in the hard sphere model; a bounded convex domain with
a flat boundary for $p = 2$, and a small ball for $p = 3$.
