Entropy inequalities are very important in information theory and they play a crucial role in various communication-theoretic problems, for example, in the study of the degrees of freedom of interference channels. In this thesis, we are concerned with the additive-combinatorial entropy inequalities which are motivated by their combinatorial counterparts: cardinality inequalities for subsets of abelian groups. As opposed to the existing approaches in the literature in the study of the discrete and continuous entropy inequalities, we consider a general framework of balanced linear entropy inequalities. In particular, we show a fundamental equivalence relationship between these classes of discrete and continuous entropy inequalities. In other words, we show that a balanced Shannon entropy inequality holds if and only if the corresponding differential entropy inequality holds. We also investigate these findings in a more general setting of connected abelian Lie groups and in the study of the sharp constants for entropy inequalities
