  In this paper, we consider the concept of $C$-star body in a fixed pointed
closed convex cone $C$ and study the dual mixed volume for $C$-star bodies. For
$C$-star bodies, we establish the corresponding dual Brunn-Minkowski
inequality, the dual Minkowski inequality and the dual Aleksandrov-Fenchel
inequality. Our dual Brunn-Minkowski inequality for $C$-star bodies strengthens
Schneider's Brunn-Minkowski inequality for $C$-coconvex sets.
