We study structured mixed integer sets with a regular cone such as nonnegative orthant, Lorentz cone or positive semidefinite cone. In a unified framework, we introduce minimal inequalities and show that under mild assumptions, minimal inequalities together with the trivial conic inequalities provide complete convex hull description. We also provide a characterization of minimal inequalities by establishing necessary conditions for an inequality to be minimal. This characterization leads to a more general class of generalized subadditive inequalities, which includes minimal inequalities as a subclass. We establish relations between generalized subadditive inequalities and the support functions of sets with certain structure. Our framework generalizes the results from the mixed integer linear case, so whenever possible we highlight the connections to the existing literature.