  In this paper, combining the covolume, we study the Minkowski theory for the
non-compact convex set with an asymptotic boundary condition. In particular,
the mixed covolume of two non-compact convex sets is introduced and its
geometric interpretation is obtained by the Hadamard variational formula. The
Brunn-Minkowski and Minkowski inequalities for covolume are established, and
the equivalence of these two inequalities are discussed as well. The Minkowski
problem for non-compact convex set is proposed and solved under the asymptotic
conditions. In the end, we give a solution to the Minkowski problem for
$\sigma$-finite measure on the conic domain $\Omega_C$.
